##### Class 9^{th} Mathematics Term 3 Tamilnadu Board Solution

**Exercise 2.1**- 749300000000 Represent the following numbers in the scientific notation.…
- 13000000 Represent the following numbers in the scientific notation.…
- 105003 Represent the following numbers in the scientific notation.…
- 543600000000000 Represent the following numbers in the scientific notation.…
- 0.0096 Represent the following numbers in the scientific notation.…
- 0.0000013307 Represent the following numbers in the scientific notation.…
- 0.0000000022 Represent the following numbers in the scientific notation.…
- 0.0000000000009 Represent the following numbers in the scientific notation.…
- 3.25 × 10-6 Write the following numbers in decimal form.
- 4.134 × 10-4 Write the following numbers in decimal form.
- 4.134 × 10^4 Write the following numbers in decimal form.
- 1.86 × 10^7 Write the following numbers in decimal form.
- 9.87 × 10^9 Write the following numbers in decimal form.
- 1.432 × 10-9 Write the following numbers in decimal form.
- (1000)^2 × (20)^6 Represent the following numbers in scientific notation.…
- (1500)^3 (0.0001)^2 Represent the following numbers in scientific notation.…
- (16000)^3 ÷ (200)^4 Represent the following numbers in scientific notation.…
- (0.003)^7 (0.0002)^5 ÷ (0.001)^3 Represent the following numbers in scientific…
- (11000)^3 (0.003)^2 ÷ (30000) Represent the following numbers in scientific…

**Exercise 2.2**- State whether each of the following statements is true or false. (i) log5125 = 3…
- 2^4 = 16 Obtain the equivalent logarithmic form of the following.…
- 3^5 = 243 Obtain the equivalent logarithmic form of the following.…
- 10-1 = 0.1 Obtain the equivalent logarithmic form of the following.…
- 8^- 2/3 = 1/4 Obtain the equivalent logarithmic form of the following.…
- 25^1/2 = 5 Obtain the equivalent logarithmic form of the following.…
- 12^-2 = 1/144 Obtain the equivalent logarithmic form of the following.…
- log6216 = 3 Obtain the equivalent exponential form of the following.…
- log_93 = 1/2 Obtain the equivalent exponential form of the following.…
- log51 = 0 Obtain the equivalent exponential form of the following.…
- log_ root 3 9 = 4 Obtain the equivalent exponential form of the following.…
- log_64 (1/8) = - 1/2 Obtain the equivalent exponential form of the following.…
- log0.58 = - 3 Obtain the equivalent exponential form of the following.…
- log_3 (1/81) Find the value of the following
- log7 343 Find the value of the following
- log66^5 Find the value of the following
- log_ 1/2 8 Find the value of the following
- log10 0.0001 Find the value of the following
- log_ root 3 9 root 3 Find the value of the following
- log_2x = 1/2 Solve the following equations.
- log_ 1/2 x = 3 Solve the following equations.
- log3 y = - 2 Solve the following equations.
- log_x125 root 5 = 7 Solve the following equations.
- logx 0.001 = - 3 Solve the following equations.
- x + 2 log27 9 = 0 Solve the following equations.
- log103 + log103 Simplify the following.
- log2535 - log2510 Simplify the following.
- log721 + log777 + log788 - log7121 - log724 Simplify the following.…
- log_816+log_852 - 1/log_138 Simplify the following.
- 5log102 + 2log103 - 6log644 Simplify the following.
- log108 + log105 - log104 Simplify the following.
- log4(x + 4) + log48 = 2 Solve the equation in each of the following.…
- log6(x + 4) - log6(x - 1) = 2 Solve the equation in each of the following.…
- log_2x+log_4x+log_8x = 11/6 Solve the equation in each of the following.…
- log4(8log2x) = 2 Solve the equation in each of the following.
- log105 + log10(5x + 1) = log10(x + 5) + 1 Solve the equation in each of the…
- 4log2x - log25 = log2125 Solve the equation in each of the following.…
- log325 + log3x = 3log35 Solve the equation in each of the following.…
- log_3 (root 5x-2) - 1/2 = log_3 (root x+4) Solve the equation in each of the…
- Given loga2 = x, loga 3 = y and loga 5 = z. Find the value in each of the…
- log101600 = 2 + 4log102 Prove the following equations.
- log1012500 = 2 + 3log105 Prove the following equations.
- log102500 = 4 - 2log102 Prove the following equations.
- log100.16 = 2log104 - 2 Prove the following equations.
- log50.00125 = 3 - 5log510 Prove the following equations.
- log_51875 = 1/2 log_536 - 1/3 log_58+20log_322 Prove the following equations.…

**Exercise 2.3**- 92.43 Write each of the following in scientific notation:
- 0.9243 Write each of the following in scientific notation:
- 9243 Write each of the following in scientific notation:
- 924300 Write each of the following in scientific notation:
- 0.009243 Write each of the following in scientific notation:
- 0.09243 Write each of the following in scientific notation:
- log 4576 Write the characteristic of each of the following
- log 24.56 Write the characteristic of each of the following
- log 0.00257 Write the characteristic of each of the following
- log 0.0756 Write the characteristic of each of the following
- log 0.2798 Write the characteristic of each of the following
- log 6.453 Write the characteristic of each of the following
- log 23750 The mantissa of log 23750 is 0.3756. Find the value of the following.…
- log 23.75 The mantissa of log 23750 is 0.3756. Find the value of the following.…
- log 2.375 The mantissa of log 23750 is 0.3756. Find the value of the following.…
- log 0.2375 The mantissa of log 23750 is 0.3756. Find the value of the…
- log 23750000 The mantissa of log 23750 is 0.3756. Find the value of the…
- log 0.00002375 The mantissa of log 23750 is 0.3756. Find the value of the…
- log 23.17 Using logarithmic table find the value of the following.…
- log 9.321 Using logarithmic table find the value of the following.…
- log 329.5 Using logarithmic table find the value of the following.…
- log 0.001364 Using logarithmic table find the value of the following.…
- log 0.9876 Using logarithmic table find the value of the following.…
- log 6576 Using logarithmic table find the value of the following.…
- Using antilogarithmic table find the value of the following. i. antilog 3.072…
- 816.3 × 37.42 Evaluate:
- 816.3 ÷ 37.42 Evaluate:
- 0.000645 × 82.3 Evaluate:
- 0.3421 ÷ 0.09782 Evaluate:
- (50.49)^5 Evaluate:
- cube root 561.4 Evaluate:
- 175.23 x 22.159/1828.56 Evaluate:
- cube root 28 x root [5]729/root 46.35 Evaluate:
- (76.25)^3 x cube root 1.928/(42.75)^5 x 0.04623 Evaluate:
- cube root 0.7214 x 20.37/69.8 Evaluate:
- log9 63.28 Evaluate:
- log3 7 Evaluate:

**Exercise 2.4**- Convert 4510 to base 2
- Convert 7310 to base 2.
- Convert 11010112 to base 10.
- Convert 1112 to base 10.
- Convert 98710 to base 5.
- Convert 123810 to base 5.
- Convert 102345 to base 10.
- Convert 2114235 to base 10.
- Convert 9856710 to base 8.
- Convert 68810 to base 8.
- Convert 471568 to base 10.
- Convert 58510 to base 2,5 and 8.

**Exercise 2.5**- The scientific notation of 923.4 isA. 9.234 × 10-2 B. 9.234 × 10^2 C. 9.234 ×…
- The scientific notation of 0.00036 isA. 3.6 × 10-3 B. 3.6 × 10^3 C. 3.6 × 10-4…
- The decimal form of 2.57 x 10^3 isA. 257 B. 2570 C. 25700 D. 257000…
- The decimal form of 3.506 × 10-2 isA. 0.03506 B. 0.003506 C. 35.06 D. 350.6…
- The logarithmic form of 5^2 = 25 isA. log52 = 25 B. log25 = 25 C. log525 =2 D.…
- The exponential form of log216 = 4 isA. 2^4 = 16 B. 4^2 = 16 C. 2^16 = 4 D. 4^16…
- The value of log_ 3/4 (4/3) isA. - 2 B. 1 C. 2 D. - 1
- The value of log_497 isA. 2 B. 1/2 C. 1/7 D. 1
- The value of log_ 1/2 4 isA. - 2 B. 0 C. 1/2 D. 2
- log108 + log105- log104 =A. log109 B. log1036 C. 1 D. - 1

**Exercise 2.1**

- 749300000000 Represent the following numbers in the scientific notation.…
- 13000000 Represent the following numbers in the scientific notation.…
- 105003 Represent the following numbers in the scientific notation.…
- 543600000000000 Represent the following numbers in the scientific notation.…
- 0.0096 Represent the following numbers in the scientific notation.…
- 0.0000013307 Represent the following numbers in the scientific notation.…
- 0.0000000022 Represent the following numbers in the scientific notation.…
- 0.0000000000009 Represent the following numbers in the scientific notation.…
- 3.25 × 10-6 Write the following numbers in decimal form.
- 4.134 × 10-4 Write the following numbers in decimal form.
- 4.134 × 10^4 Write the following numbers in decimal form.
- 1.86 × 10^7 Write the following numbers in decimal form.
- 9.87 × 10^9 Write the following numbers in decimal form.
- 1.432 × 10-9 Write the following numbers in decimal form.
- (1000)^2 × (20)^6 Represent the following numbers in scientific notation.…
- (1500)^3 (0.0001)^2 Represent the following numbers in scientific notation.…
- (16000)^3 ÷ (200)^4 Represent the following numbers in scientific notation.…
- (0.003)^7 (0.0002)^5 ÷ (0.001)^3 Represent the following numbers in scientific…
- (11000)^3 (0.003)^2 ÷ (30000) Represent the following numbers in scientific…

**Exercise 2.2**

- State whether each of the following statements is true or false. (i) log5125 = 3…
- 2^4 = 16 Obtain the equivalent logarithmic form of the following.…
- 3^5 = 243 Obtain the equivalent logarithmic form of the following.…
- 10-1 = 0.1 Obtain the equivalent logarithmic form of the following.…
- 8^- 2/3 = 1/4 Obtain the equivalent logarithmic form of the following.…
- 25^1/2 = 5 Obtain the equivalent logarithmic form of the following.…
- 12^-2 = 1/144 Obtain the equivalent logarithmic form of the following.…
- log6216 = 3 Obtain the equivalent exponential form of the following.…
- log_93 = 1/2 Obtain the equivalent exponential form of the following.…
- log51 = 0 Obtain the equivalent exponential form of the following.…
- log_ root 3 9 = 4 Obtain the equivalent exponential form of the following.…
- log_64 (1/8) = - 1/2 Obtain the equivalent exponential form of the following.…
- log0.58 = - 3 Obtain the equivalent exponential form of the following.…
- log_3 (1/81) Find the value of the following
- log7 343 Find the value of the following
- log66^5 Find the value of the following
- log_ 1/2 8 Find the value of the following
- log10 0.0001 Find the value of the following
- log_ root 3 9 root 3 Find the value of the following
- log_2x = 1/2 Solve the following equations.
- log_ 1/2 x = 3 Solve the following equations.
- log3 y = - 2 Solve the following equations.
- log_x125 root 5 = 7 Solve the following equations.
- logx 0.001 = - 3 Solve the following equations.
- x + 2 log27 9 = 0 Solve the following equations.
- log103 + log103 Simplify the following.
- log2535 - log2510 Simplify the following.
- log721 + log777 + log788 - log7121 - log724 Simplify the following.…
- log_816+log_852 - 1/log_138 Simplify the following.
- 5log102 + 2log103 - 6log644 Simplify the following.
- log108 + log105 - log104 Simplify the following.
- log4(x + 4) + log48 = 2 Solve the equation in each of the following.…
- log6(x + 4) - log6(x - 1) = 2 Solve the equation in each of the following.…
- log_2x+log_4x+log_8x = 11/6 Solve the equation in each of the following.…
- log4(8log2x) = 2 Solve the equation in each of the following.
- log105 + log10(5x + 1) = log10(x + 5) + 1 Solve the equation in each of the…
- 4log2x - log25 = log2125 Solve the equation in each of the following.…
- log325 + log3x = 3log35 Solve the equation in each of the following.…
- log_3 (root 5x-2) - 1/2 = log_3 (root x+4) Solve the equation in each of the…
- Given loga2 = x, loga 3 = y and loga 5 = z. Find the value in each of the…
- log101600 = 2 + 4log102 Prove the following equations.
- log1012500 = 2 + 3log105 Prove the following equations.
- log102500 = 4 - 2log102 Prove the following equations.
- log100.16 = 2log104 - 2 Prove the following equations.
- log50.00125 = 3 - 5log510 Prove the following equations.
- log_51875 = 1/2 log_536 - 1/3 log_58+20log_322 Prove the following equations.…

**Exercise 2.3**

- 92.43 Write each of the following in scientific notation:
- 0.9243 Write each of the following in scientific notation:
- 9243 Write each of the following in scientific notation:
- 924300 Write each of the following in scientific notation:
- 0.009243 Write each of the following in scientific notation:
- 0.09243 Write each of the following in scientific notation:
- log 4576 Write the characteristic of each of the following
- log 24.56 Write the characteristic of each of the following
- log 0.00257 Write the characteristic of each of the following
- log 0.0756 Write the characteristic of each of the following
- log 0.2798 Write the characteristic of each of the following
- log 6.453 Write the characteristic of each of the following
- log 23750 The mantissa of log 23750 is 0.3756. Find the value of the following.…
- log 23.75 The mantissa of log 23750 is 0.3756. Find the value of the following.…
- log 2.375 The mantissa of log 23750 is 0.3756. Find the value of the following.…
- log 0.2375 The mantissa of log 23750 is 0.3756. Find the value of the…
- log 23750000 The mantissa of log 23750 is 0.3756. Find the value of the…
- log 0.00002375 The mantissa of log 23750 is 0.3756. Find the value of the…
- log 23.17 Using logarithmic table find the value of the following.…
- log 9.321 Using logarithmic table find the value of the following.…
- log 329.5 Using logarithmic table find the value of the following.…
- log 0.001364 Using logarithmic table find the value of the following.…
- log 0.9876 Using logarithmic table find the value of the following.…
- log 6576 Using logarithmic table find the value of the following.…
- Using antilogarithmic table find the value of the following. i. antilog 3.072…
- 816.3 × 37.42 Evaluate:
- 816.3 ÷ 37.42 Evaluate:
- 0.000645 × 82.3 Evaluate:
- 0.3421 ÷ 0.09782 Evaluate:
- (50.49)^5 Evaluate:
- cube root 561.4 Evaluate:
- 175.23 x 22.159/1828.56 Evaluate:
- cube root 28 x root [5]729/root 46.35 Evaluate:
- (76.25)^3 x cube root 1.928/(42.75)^5 x 0.04623 Evaluate:
- cube root 0.7214 x 20.37/69.8 Evaluate:
- log9 63.28 Evaluate:
- log3 7 Evaluate:

**Exercise 2.4**

- Convert 4510 to base 2
- Convert 7310 to base 2.
- Convert 11010112 to base 10.
- Convert 1112 to base 10.
- Convert 98710 to base 5.
- Convert 123810 to base 5.
- Convert 102345 to base 10.
- Convert 2114235 to base 10.
- Convert 9856710 to base 8.
- Convert 68810 to base 8.
- Convert 471568 to base 10.
- Convert 58510 to base 2,5 and 8.

**Exercise 2.5**

- The scientific notation of 923.4 isA. 9.234 × 10-2 B. 9.234 × 10^2 C. 9.234 ×…
- The scientific notation of 0.00036 isA. 3.6 × 10-3 B. 3.6 × 10^3 C. 3.6 × 10-4…
- The decimal form of 2.57 x 10^3 isA. 257 B. 2570 C. 25700 D. 257000…
- The decimal form of 3.506 × 10-2 isA. 0.03506 B. 0.003506 C. 35.06 D. 350.6…
- The logarithmic form of 5^2 = 25 isA. log52 = 25 B. log25 = 25 C. log525 =2 D.…
- The exponential form of log216 = 4 isA. 2^4 = 16 B. 4^2 = 16 C. 2^16 = 4 D. 4^16…
- The value of log_ 3/4 (4/3) isA. - 2 B. 1 C. 2 D. - 1
- The value of log_497 isA. 2 B. 1/2 C. 1/7 D. 1
- The value of log_ 1/2 4 isA. - 2 B. 0 C. 1/2 D. 2
- log108 + log105- log104 =A. log109 B. log1036 C. 1 D. - 1

###### Exercise 2.1

**Question 1.**Represent the following numbers in the scientific notation.

749300000000

**Answer:**The given number is 7 4 9 3 0 0 0 0 0 0 0 0 . (In integers decimal point at the end is usually omitted.)

Move the decimal point so that there is only one non - zero digit to its left.

The decimal point is to be moved 11 places to the left of its original position. So, the power of 10 is 11.

(The count of the number of digits between the old and new decimal point gives n the power of 10.)

Therefore, scientific notation is 7.49300000000×10^{11} = 7.493×10^{11}.

**Question 2.**Represent the following numbers in the scientific notation.

13000000

**Answer:**The given number is 1 3 0 0 0 0 0 0 .

The decimal point is to be moved 7 places to the left of its original position. So the power of 10 is 7.

Therefore, scientific notation is 1.3000000×10^{7} = 1.3×10^{7}

**Question 3.**Represent the following numbers in the scientific notation.

105003

**Answer:**The given number is 1 0 5 0 0 3 .

The decimal point is to be moved 5 places to the left of its original position. So the power of 10 is 5.

Therefore,scientific notation is 1.05003×10^{5}

**Question 4.**Represent the following numbers in the scientific notation.

543600000000000

**Answer:**The given number is 5 4 3 6 0 0 0 0 0 0 0 0 0 0 0 .

The decimal point is to be moved 14 places to the left of its original position. So the power of 10 is 14.

Therefore,scientific notation is 5.436×10^{14}.

**Question 5.**Represent the following numbers in the scientific notation.

0.0096

**Answer:**The given number is 0 . 0 0 9 6

The decimal point is to be moved 3 places to the right of its original position. So the power of 10 is - 3.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore,scientific notation is 9.6×10 ^{- 3}

**Question 6.**Represent the following numbers in the scientific notation.

0.0000013307

**Answer:**The given number is 0 . 0 0 0 0 0 1 3 3 0 7

The decimal point is to be moved 6 places to the right of its original position. So the power of 10 is - 6.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore, scientific notation is 1.3307×10 ^{- 6}

**Question 7.**Represent the following numbers in the scientific notation.

0.0000000022

**Answer:**The given number is 0 . 0 0 0 0 0 0 0 0 2 2

The decimal point is to be moved 9 places to the right of its original position. So the power of 10 is - 9.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore, scientific notation is 2.2×10 ^{- 9}

**Question 8.**Represent the following numbers in the scientific notation.

0.0000000000009

**Answer:**The given number is 0 . 0 0 0 0 0 0 0 0 0 0 0 0 9

The decimal point is to be moved 13 places to the right of its original position. So the power of 10 is - 13.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore, scientific notation is 9.0×10 ^{- 13}

**Question 9.**Write the following numbers in decimal form.

3.25 × 10^{-6}

**Answer:**The given number is 3.25 × 10^{-6}.

In this number the decimal number is 3.25

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is - 6.

So, the number in decimal form is 0.00000325

**Question 10.**Write the following numbers in decimal form.

4.134 × 10^{-4}

**Answer:**The given number is 4.134 × 10^{-4}

In this number the decimal number is 4.134

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is - 4.

So, the number in decimal form is 0.0004134

**Question 11.**Write the following numbers in decimal form.

4.134 × 10^{4}

**Answer:**In decimal form, the given expression is written as:

4.134 × 10^{4}

= 41.34 × 10^{3}

= 413.4 × 10^{2}

= 4134 × 10^{1}

= 41340

Hence, the decimal form of the given expression is: 41340

**Question 12.**Write the following numbers in decimal form.

1.86 × 10^{7}

**Answer:**The given number is 1.86×10^{7}.

In this number the decimal number is 1.86

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is 7.

So, the number in becomes 18600000.00.

Therefore, the number in decimal form is 18600000.

**Question 13.**Write the following numbers in decimal form.

9.87 × 10^{9}

**Answer:**The given number is 9.87×10^{9}

In this number the decimal number is 9.87

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is 9.

So, the number in becomes 9870000000.00

Therefore,the number in decimal form is 9870000000.

**Question 14.**Write the following numbers in decimal form.

1.432 × 10^{-9}

**Answer:**The given number is 1.432×10^{-9}

In this number the decimal number is 1.432

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is 9.

So, the number indecimal form is 0.000000001432

**Question 15.**Represent the following numbers in scientific notation.

(1000)^{2} × (20)^{6}

**Answer:**In scientific notation,

1000 = (1.0×10^{3}) and 20 = (2.0×10^{1})^{6}

∴(1000)^{2}×(20)^{6} = (1.0×10^{3})^{2}×(2.0×10^{1})^{6}

= (1.0)^{2}×(10^{3})^{2}×(2.0)^{6}×(10^{1})^{6}

= 1×10^{6}×64×10^{6}

= 64×10^{12}

= 6.4×10^{1}×10^{12}

= 6.4×10^{13}

∴ (1000)^{2} x (20)^{6} in scientific notation is 6.4×10^{13}

**Question 16.**Represent the following numbers in scientific notation.

(1500)^{3}(0.0001)^{2}

**Answer:**In scientific notation,

1500 = (1.5×10^{3}) and 0.0001 = (1.0×10 ^{- 4})

∴(1500)^{3}×(0.0001)^{2} = (1.5×10^{3})^{3}×(1.0×10 ^{- 4})^{2}

= (1.5)^{3}×(10^{3})^{3}×(1.0)^{2}×(10 ^{- 4})^{2}

= 3.375 ×(10)^{9}×1×(10) ^{- 8}

= 3.375×(10)^{1}

∴ (1500)^{3}×(0.0001)^{2} in scientific notation is 3.375×10^{1}

**Question 17.**Represent the following numbers in scientific notation.

(16000)^{3} ÷ (200)^{4}

**Answer:**In scientific notation,

16000 = (1.6×10^{3}) and 200 = (2.0×10^{2})

∴ (16000)^{3} (200)^{4} = (1.6×10^{4})^{3} ÷ (2.0×10^{2})^{4}

∴ (16000)^{3} (200)^{4} in scientific notation is 2.56 ×103

**Question 18.**Represent the following numbers in scientific notation.

(0.003)^{7}(0.0002)^{5} ÷ ( 0.001)^{3}

**Answer:**In scientific notation,

0.003 = (3.0)×(10) ^{- 3}

0.0002 = (2.0)×(10) ^{- 4}

0.001 = (1.0)×(10) ^{- 3}

∴

⇒

= 6.9984×10 ^{- 28}

∴ (0.003)^{7}(0.0002)^{5} ÷ (0.001)^{3} in scientific notation is 6.9984×10 - 28

**Question 19.**Represent the following numbers in scientific notation.

(11000)^{3} (0.003)^{2} ÷ (30000)

**Answer:**(11000)^{3} (0.003)^{2}( 30000)

__Explanation:__ In scientific notation,

11000 = (1.1)×(10)^{4}

0.003 = (3.0)×(10) ^{- 3}

30000 = (3.0)×(10)^{5}

∴ (11000)^{3} (0.003)^{2}( 30000)

⇒

1.331×10^{6}×3×10 ^{- 5}

= 3.993×10^{1}

∴ (11000)^{3} (0.003)^{2} ÷ (3000) in scientific notation is 3.993×10^{1}

**Question 1.**

Represent the following numbers in the scientific notation.

749300000000

**Answer:**

The given number is 7 4 9 3 0 0 0 0 0 0 0 0 . (In integers decimal point at the end is usually omitted.)

Move the decimal point so that there is only one non - zero digit to its left.

The decimal point is to be moved 11 places to the left of its original position. So, the power of 10 is 11.

(The count of the number of digits between the old and new decimal point gives n the power of 10.)

Therefore, scientific notation is 7.49300000000×10^{11} = 7.493×10^{11}.

**Question 2.**

Represent the following numbers in the scientific notation.

13000000

**Answer:**

The given number is 1 3 0 0 0 0 0 0 .

The decimal point is to be moved 7 places to the left of its original position. So the power of 10 is 7.

Therefore, scientific notation is 1.3000000×10^{7} = 1.3×10^{7}

**Question 3.**

Represent the following numbers in the scientific notation.

105003

**Answer:**

The given number is 1 0 5 0 0 3 .

The decimal point is to be moved 5 places to the left of its original position. So the power of 10 is 5.

Therefore,scientific notation is 1.05003×10^{5}

**Question 4.**

Represent the following numbers in the scientific notation.

543600000000000

**Answer:**

The given number is 5 4 3 6 0 0 0 0 0 0 0 0 0 0 0 .

The decimal point is to be moved 14 places to the left of its original position. So the power of 10 is 14.

Therefore,scientific notation is 5.436×10^{14}.

**Question 5.**

Represent the following numbers in the scientific notation.

0.0096

**Answer:**

The given number is 0 . 0 0 9 6

The decimal point is to be moved 3 places to the right of its original position. So the power of 10 is - 3.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore,scientific notation is 9.6×10 ^{- 3}

**Question 6.**

Represent the following numbers in the scientific notation.

0.0000013307

**Answer:**

The given number is 0 . 0 0 0 0 0 1 3 3 0 7

The decimal point is to be moved 6 places to the right of its original position. So the power of 10 is - 6.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore, scientific notation is 1.3307×10 ^{- 6}

**Question 7.**

Represent the following numbers in the scientific notation.

0.0000000022

**Answer:**

The given number is 0 . 0 0 0 0 0 0 0 0 2 2

The decimal point is to be moved 9 places to the right of its original position. So the power of 10 is - 9.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore, scientific notation is 2.2×10 ^{- 9}

**Question 8.**

Represent the following numbers in the scientific notation.

0.0000000000009

**Answer:**

The given number is 0 . 0 0 0 0 0 0 0 0 0 0 0 0 9

The decimal point is to be moved 13 places to the right of its original position. So the power of 10 is - 13.(If the decimal is shifted to the right ,the exponent n is negative.)

Therefore, scientific notation is 9.0×10 ^{- 13}

**Question 9.**

Write the following numbers in decimal form.

3.25 × 10^{-6}

**Answer:**

The given number is 3.25 × 10^{-6}.

In this number the decimal number is 3.25

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is - 6.

So, the number in decimal form is 0.00000325

**Question 10.**

Write the following numbers in decimal form.

4.134 × 10^{-4}

**Answer:**

The given number is 4.134 × 10^{-4}

In this number the decimal number is 4.134

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is - 4.

So, the number in decimal form is 0.0004134

**Question 11.**

Write the following numbers in decimal form.

4.134 × 10^{4}

**Answer:**

In decimal form, the given expression is written as:

4.134 × 10^{4}

= 41.34 × 10^{3}

= 413.4 × 10^{2}

= 4134 × 10^{1}

= 41340

Hence, the decimal form of the given expression is: 41340

**Question 12.**

Write the following numbers in decimal form.

1.86 × 10^{7}

**Answer:**

The given number is 1.86×10^{7}.

In this number the decimal number is 1.86

Now we have to move the decimal point the number of places specified by the power of 10:to the right if positive to the left if negative. Add zeros if necessary. Rewrite the number in decimal form.

Here power of 10 i.e. n is 7.

So, the number in becomes 18600000.00.

Therefore, the number in decimal form is 18600000.

**Question 13.**

Write the following numbers in decimal form.

9.87 × 10^{9}

**Answer:**

The given number is 9.87×10^{9}

In this number the decimal number is 9.87

Here power of 10 i.e. n is 9.

So, the number in becomes 9870000000.00

Therefore,the number in decimal form is 9870000000.

**Question 14.**

Write the following numbers in decimal form.

1.432 × 10^{-9}

**Answer:**

The given number is 1.432×10^{-9}

In this number the decimal number is 1.432

Here power of 10 i.e. n is 9.

So, the number indecimal form is 0.000000001432

**Question 15.**

Represent the following numbers in scientific notation.

(1000)^{2} × (20)^{6}

**Answer:**

In scientific notation,

1000 = (1.0×10^{3}) and 20 = (2.0×10^{1})^{6}

∴(1000)^{2}×(20)^{6} = (1.0×10^{3})^{2}×(2.0×10^{1})^{6}

= (1.0)^{2}×(10^{3})^{2}×(2.0)^{6}×(10^{1})^{6}

= 1×10^{6}×64×10^{6}

= 64×10^{12}

= 6.4×10^{1}×10^{12}

= 6.4×10^{13}

∴ (1000)^{2} x (20)^{6} in scientific notation is 6.4×10^{13}

**Question 16.**

Represent the following numbers in scientific notation.

(1500)^{3}(0.0001)^{2}

**Answer:**

In scientific notation,

1500 = (1.5×10^{3}) and 0.0001 = (1.0×10 ^{- 4})

∴(1500)^{3}×(0.0001)^{2} = (1.5×10^{3})^{3}×(1.0×10 ^{- 4})^{2}

= (1.5)^{3}×(10^{3})^{3}×(1.0)^{2}×(10 ^{- 4})^{2}

= 3.375 ×(10)^{9}×1×(10) ^{- 8}

= 3.375×(10)^{1}

∴ (1500)^{3}×(0.0001)^{2} in scientific notation is 3.375×10^{1}

**Question 17.**

Represent the following numbers in scientific notation.

(16000)^{3} ÷ (200)^{4}

**Answer:**

In scientific notation,

16000 = (1.6×10^{3}) and 200 = (2.0×10^{2})

∴ (16000)^{3} (200)^{4} = (1.6×10^{4})^{3} ÷ (2.0×10^{2})^{4}

∴ (16000)^{3} (200)^{4} in scientific notation is 2.56 ×103

**Question 18.**

Represent the following numbers in scientific notation.

(0.003)^{7}(0.0002)^{5} ÷ ( 0.001)^{3}

**Answer:**

In scientific notation,

0.003 = (3.0)×(10) ^{- 3}

0.0002 = (2.0)×(10) ^{- 4}

0.001 = (1.0)×(10) ^{- 3}

∴

⇒

= 6.9984×10 ^{- 28}

∴ (0.003)^{7}(0.0002)^{5} ÷ (0.001)^{3} in scientific notation is 6.9984×10 - 28

**Question 19.**

Represent the following numbers in scientific notation.

(11000)^{3} (0.003)^{2} ÷ (30000)

**Answer:**

(11000)^{3} (0.003)^{2}( 30000)

__Explanation:__ In scientific notation,

11000 = (1.1)×(10)^{4}

0.003 = (3.0)×(10) ^{- 3}

30000 = (3.0)×(10)^{5}

∴ (11000)^{3} (0.003)^{2}( 30000)

⇒

1.331×10^{6}×3×10 ^{- 5}

= 3.993×10^{1}

∴ (11000)^{3} (0.003)^{2} ÷ (3000) in scientific notation is 3.993×10^{1}

###### Exercise 2.2

**Question 1.**State whether each of the following statements is true or false.

(i) log_{5}125 = 3

(ii)

(iii) log_{4}(6 + 3) = log_{4}6 + log_{4}3

(iv)

(v)

(vi) log_{a}M - N = log_{a}Mlog_{a}N

**Answer:**(i) True

log_{5}125 = 3

⇒ 5^{3} = 125

(∵ x = log_{a}b is the logarithmic form of the exponential form a^{x} = b)

This is true.

(ii) False

⇒

(∵ x = log_{a}b is the logarithmic form of the exponential form ax = b)

Here

Therefore, this False.

(iii) False

Here its given log_{4}(6 + 3) = log_{4}6 + log_{4}3

Let us consider the RHS i.e.

log_{4}6 + log_{4}3 = log_{4}(6×3) (∵ according to the product rule loga(M×N) = logaM + logaN;

a,M,N are positive numbers,a≠1)

But here LHS is log4 (6 + 3)

Hence it’s False.

(iv) False

Here it’s given

Let us consider the LHS i.e.

(∵ log_{a}M ÷ log_{a}N = log_{a}M - log_{a}N

;a,M,N are positive numbers ,a≠1)

But here the RHS is

Hence both the sides are not equal and therefore it’s False.

(v) True

Here it’s given:

⇒ (∵ x = log_{a}b is the logarithmic form of the exponential form a^{x} = b)

⇒

Hence LHS = RHS

Therefore this is True.

(vi) False

Here it’s given that log_{a} (M - N) = log_{a} M ÷ log_{a}N

Let us consider the RHS

log_{a}M ÷ log_{a}N = log_{a}M - log_{a}N

(∵ according to quotient rule,log_{a}M ÷ log_{a}N = log_{a}M - log_{a}N ;a,M,N are positive numbers,a≠1)

But the LHS is log_{a}(M - N)

Therefore LHS≠RHS

Hence it’s False.

**Question 2.**Obtain the equivalent logarithmic form of the following.

2^{4} = 16

**Answer:**Here it’s given that 2^{4} = 16,

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the equation 2^{4} = 16 (a = 2,b = 16 ,x = 4)

⇒ log_{2}16 = 4

**Question 3.**Obtain the equivalent logarithmic form of the following.

3^{5} = 243

**Answer:**Here it’s given that 3^{5} = 243

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the equation 3^{5} = 243 (a = 3,b = 343 ,x = 5)

⇒ log_{3}243 = 5

**Question 4.**Obtain the equivalent logarithmic form of the following.

10^{-1} = 0.1

**Answer:**Here it’s given that 10 ^{- 1} = 0.1

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the equation 10 ^{- 1} = 0.1 (a = 10,b = 0.1,x = - 1)

⇒

**Question 5.**Obtain the equivalent logarithmic form of the following.

**Answer:**Here it’s given that

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the given equation (a = 8,,)

⇒

**Question 6.**Obtain the equivalent logarithmic form of the following.

**Answer:**Here it’s given that

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the given equation (a = 25,b = 5,)

⇒

**Question 7.**Obtain the equivalent logarithmic form of the following.

**Answer:**Here it’s given that

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the equation (a = 12,,x = - 2)

⇒

**Question 8.**Obtain the equivalent exponential form of the following.

log_{6}216 = 3

**Answer:**Here it’s given that log_{6}216 = 3

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation log_{6}216 = 3 ( a = 6,b = 216 ,x = 3)

⇒ 6^{3} = 216

**Question 9.**Obtain the equivalent exponential form of the following.

**Answer:**Here it’s given that

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation ( a = 9,b = 3 ,)

⇒

**Question 10.**Obtain the equivalent exponential form of the following.

log_{5}1 = 0

**Answer:**Here it’s given that log_{5}1 = 0

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation log_{5}1 = 0 (a = 5,b = 1,x = 0)

⇒ 5^{0} = 1

**Question 11.**Obtain the equivalent exponential form of the following.

**Answer:**Here it’s given that

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation (,b = 9,x = 4)

⇒

**Question 12.**Obtain the equivalent exponential form of the following.

**Answer:**Here it’s given that

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation ( a = 64,,)

⇒

**Question 13.**Obtain the equivalent exponential form of the following.

log_{0.5}8 = - 3

**Answer:**Here it’s given that log_{0.5}8 = - 3

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation log_{0.5}8 = - 3 (a = 0.5,b = 8,x = - 3)

⇒ (0.5) ^{- 3} = 8

**Question 14.**Find the value of the following

**Answer:**

i.e. log_{3}(3 ^{- 4}) = - 4(log_{3}3)

(∵ nlog_{a}M = log_{a}M^{n})

⇒ - 4(1) = - 4

(log_{a}a = 1)

**Question 15.**Find the value of the following

log_{7} 343

**Answer:**log_{7}343 = log_{7}7^{3}

⇒ 3log_{7}7 (∵ nlog_{a}M = log_{a}M^{n})

⇒ 1(∵ log_{a}a = 1)

**Question 16.**Find the value of the following

log_{6}6^{5}

**Answer:**log_{6}6^{5}

⇒ 5log_{6}6

(∵ nlog_{a}M = log_{a}M^{n})

= 5(1)

(∵ log_{a}a = 1)

= 5

**Question 17.**Find the value of the following

**Answer:**Here we have i.e.

⇒ , here is

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

⇒ 3( - 1) = - 3

**Question 18.**Find the value of the following

log_{10} 0.0001

**Answer:**Here we have log100.0001, i.e.

⇒

⇒ - 4log_{10}10 (∵ nlog_{a}M = log_{a}M^{n})

⇒ - 4(1) = - 4 (∵ log_{a}a = 1)

**Question 19.**Find the value of the following

**Answer:**Here we have ,

⇒

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

⇒

⇒

⇒ x = 5

Hence the value of is 5.

**Question 20.**Solve the following equations.

**Answer:**

⇒ i.e

**Question 21.**Solve the following equations.

**Answer:**

⇒

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

Or

Or

**Question 22.**Solve the following equations.

log_{3} y = – 2

**Answer:**log_{3}y = - 2

log_{3}y = - 2

⇒ 3 ^{- 2} = y

⇒ y = 3 ^{- 2}

⇒

i.e.

**Question 23.**Solve the following equations.

**Answer:**

⇒

⇒

⇒

∴

**Question 24.**Solve the following equations.

log_{x} 0.001 = – 3

**Answer:**log_{x}0.001 = - 3

⇒ x ^{- 3} = 0.001

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

⇒

⇒

⇒ x = 10

**Question 25.**Solve the following equations.

x + 2 log_{27} 9 = 0

**Answer:**x + 2log_{27}9 = 0

⇒ x = - 2log_{27}9

⇒ x = log_{27}9 ^{- 2}

⇒ x = log_{3}^{3}(3^{2}) ^{- 2}

⇒ x = log_{3}^{3}(3) ^{- 4}

⇒ (3^{3})^{x} = 3 ^{– 4}

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

⇒ 3x = - 4 (compare the exponents)

⇒

**Question 26.**Simplify the following.

log_{10}3 + log_{10}3

**Answer:**log_{10}3 + log_{10}3 = log_{10}(3×3) = log_{10}9

(∵ using the product rule,log_{a}(M×N) = (log_{a}M) + (log_{a}N);a,M,N are positive numbers ,a≠1)

**Question 27.**Simplify the following.

log_{25}35 – log_{25}10

**Answer:**

(using the quotient rule log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N) );a,M,N are positive numbers ,a≠1)

=

**Question 28.**Simplify the following.

log_{7}21 + log_{7}77 + log_{7}88 – log_{7}121 – log_{7}24

**Answer:**log_{7}21 + log_{7}77 + log_{7}88 - log_{7}121 - log_{7}24

⇒

(using the product rule and the quotient rule i.e.

log_{a}(M×N) = (log_{a}M) + (log_{a}N) and

log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

⇒

⇒

⇒ log_{7}7^{2}

⇒ 2log_{7}7 = 2

(∵ log_{7}7 = 1 )

**Question 29.**Simplify the following.

**Answer:**

⇒ log_{8}(16×52) - log_{8}13

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N) and )

⇒

(∵ loga(M ÷ N) = (log_{a}M) - (log_{a}N))

⇒ log_{8}(16×4) = log_{8}64

⇒ 8x = 64 or x = 2

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

**Question 30.**Simplify the following.

5log_{10}2 + 2log_{10}3 - 6log_{64}4

**Answer:**5log_{10}2 + 2log_{10}3 - 6log_{64}4

Here log_{64}4 = x

⇒ 64^{x} = 4

⇒ (4^{3})^{x} = 4

⇒ 3x = 1

⇒

∴

∴ 5log_{10}2 + 2log_{10}3 - 6log_{64}4 = log_{10}2^{5} + log_{10}3^{2} - 2

= log_{10}32 + log_{10}9 - 2log_{10}10

= log_{10}32 + log_{10}9 - log_{10}10^{2}

^{=}

**Question 31.**Simplify the following.

log_{10}8 + log_{10}5 - log_{10}4

**Answer:**log_{10}8 + log_{10}5 - log_{10}4

⇒ log_{10}(8×5) - log_{10}4

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

⇒

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

⇒ log10(2×5) = log1010 = 1

(∵ log_{a}a = 1)

**Question 32.**Solve the equation in each of the following.

log_{4}(x + 4) + log_{4}8 = 2

**Answer:**log_{4}(x + 4) + log_{4}8 = 2

⇒ log_{4}((x + 4)×8) = 2

⇒ log_{4}(8x + 32) = 2

⇒ 8x + 32 = 4^{2}

⇒ 8x + 32 = 16

⇒ 8x = 16 - 32 = - 16

⇒ 8x = - 16

⇒ x = - 2

**Question 33.**Solve the equation in each of the following.

log_{6}(x + 4) - log_{6}(x - 1) = 2

**Answer:**log_{6}(x + 4) - log_{6}(x - 1) = 2

⇒

⇒ (x + 4)(x - 1) = 6^{2} = 6×6

⇒ x + 4 = 6

⇒ x = 6 - 4 = 2

**Question 34.**Solve the equation in each of the following.

**Answer:**log_{2}x + log_{4}x + log_{8}x =

Here LHS is

⇒ log_{2}x + log_{2}^{2}x + log_{2}^{3}x

⇒

(∵)

⇒

(∵ log_{a}M^{n} = nlog_{a}M)

⇒

⇒

⇒

Now we equate LHS to the RHS i.e.

⇒

⇒ logx2 = 1 or x1 = 2 or x = 2

**Question 35.**Solve the equation in each of the following.

log_{4}(8log_{2}x) = 2

**Answer:**log_{4}(8log_{2}x) = 2

⇒ 8log_{2}x = 4^{2}

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

⇒ log_{2}x^{8} = 16

(∵ log_{a}M^{n} = nlog_{a}M)

⇒ 2^{16} = x^{8}

⇒ (2^{2})^{8} = x^{8}

⇒ x = 2^{2} = 4

**Question 36.**Solve the equation in each of the following.

log_{10}5 + log_{10}(5x + 1) = log_{10}(x + 5) + 1

**Answer:**log_{10}5 + log_{10}(5x + 1) = log_{10}(x + 5) + 1

⇒ log_{10}(5(5x + 1)) - log_{10}(x + 5) = 1

⇒

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

⇒

⇒ 25x + 5 = 10(x + 5)

⇒ 25x + 5 = 10x + 50

⇒ 25x - 10x = 50 - 5 = 45

⇒ 15x = 45

⇒ x = 3

**Question 37.**Solve the equation in each of the following.

4log_{2}x - log_{2}5 = log_{2}125

**Answer:**4log_{2}x - log_{2}5 = log_{2}125

⇒ log_{2}x^{4} - log_{2}5 = log_{2}125

⇒

⇒

⇒ x^{4} = 5×125 = 5×5^{3} = 5^{4}

⇒ x = 5

**Question 38.**Solve the equation in each of the following.

log_{3}25 + log_{3}x = 3log_{3}5

**Answer:**log_{3}25 + log_{3}x = 3log_{3}5

⇒ log_{3}(25×x) = 3log_{3}5

⇒ log_{3}(25x) = log_{3}5^{3}

⇒ 25x = 5^{3} or (5^{2})x = 5^{3}

⇒ x = 5

**Question 39.**Solve the equation in each of the following.

**Answer:**

⇒

⇒

(∵ log_{a}(M ÷ N) = log_{a}M - log_{a}N)

⇒

(∵ a^{x} = b is the exponential form of logarithmic form log_{a}b)

⇒

⇒

⇒ 5x - 2 = 3(x + 4)

⇒ 5x - 2 = 3x + 12

⇒ 5x - 3x = 12 + 2

⇒ 2x = 14

⇒ x = 7

**Question 40.**Given log_{a}2 = x, log_{a} 3 = y and log_{a} 5 = z. Find the value in each of the following in terms of x, y and z.

(i) log_{a}15 (ii) log_{a}8 (iii) log_{a}30

(iv) (v) (vi) log_{a}1.5

**Answer:**(i) log_{a}15 = log_{a}(5×3)

i.e. log_{a}(5×3) = log_{a}5 + log_{a}3

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

= z + y(∵ log_{a}5 = z,log_{a}3 = y)

(ii) log_{a}8 = log_{a}2^{3} = 3log_{a}2 = 3x

(∵ log_{a}2 = x)

(iii) log_{a}30 = log_{a}(5×3×2) = log_{a}(5) + log_{a}(3) + log_{a}(2)

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

= z + y + x

(∵ log_{a}5 = z,log_{a}3 = y,log_{a}2 = x)

= x + y + z

(iv)

⇒ log_{a}(3×3×3) - log_{a}(5×5×5)

⇒ (log_{a}3 + log_{a}3 + log_{a}3) - (log_{a}5 + log_{a}5 + log_{a}5)

⇒ (y + y + y) - (z + z + z) = 3y - 3z = 3(y - z)

(v)

⇒ log_{a}10 - log_{a}3

(∵ log_{a}(M ÷ N) = log_{a}M - log_{a}N)

Here log_{a}10 = log_{a}(5×2)

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

= log_{a}5 + log_{a}2 = z + x (∵ log_{a}5 = z,log_{a}2 = x)

(vi)

⇒

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

= y - x(∵ log_{a}3 = y,log_{a}2 = x)

**Question 41.**Prove the following equations.

log_{10}1600 = 2 + 4log_{10}2

**Answer:**log_{10}1600 = 2 + 4log_{10}2 = 2log_{10}10 + 4log_{10}2

Let us consider the RHS:

i.e. 2 + 4log_{10}2 = 2log_{10}10 + 4log_{10}2

(∵ log_{a}a = 1)

= log_{10}10^{2} + log_{10}2^{4}

(∵ log_{a}M^{n} = nlog_{a}M)

= log_{10}100 + log_{10}16

= log_{10}(100×16)

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

= log_{10}1600

Hence LHS = RHS

**Question 42.**Prove the following equations.

log_{10}12500 = 2 + 3log_{10}5

**Answer:**log_{10}12500 = 2 + 3log_{10}5 = 2log_{10}10 + 3log_{10}5

Let us consider the RHS:

i.e. 2 + 3log_{10}5 = 2log_{10}10 + 3log_{10}5

= log_{10}10^{2} + log_{10}5^{3}

(∵ log_{a}M^{n} = nlog_{a}M)

= log_{10}(10^{2}×5^{3})

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

= log_{10}(100×125)

= log_{10}(12500)

Hence LHS = RHS

**Question 43.**Prove the following equations.

log_{10}2500 = 4 - 2log_{10}2

**Answer:**log_{10}2500 = 4 - 2log_{10}2

Let us consider the RHS:

i.e. 4 - 2log_{10}2 = 4log_{10}10 - 2log_{10}2

= log_{10}10^{4} - log_{10}2^{2}

(∵ log_{a}M^{n} = nlog_{a}M)

=

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

Hence LHS = RHS

**Question 44.**Prove the following equations.

log_{10}0.16 = 2log_{10}4 – 2

**Answer:**

Let us consider the RHS:

i.e. 2log_{10}4 - 2 = 2log_{10}4 - 2log_{10}10

= log_{10}4^{2} - log_{10}10^{2}

(∵ log_{a}M^{n} = nlog_{a}M)

=

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

= log_{10}(0.16) = log_{10}0.16

Hence LHS = RHS

**Question 45.**Prove the following equations.

log_{5}0.00125 = 3 - 5log_{5}10

**Answer:**log_{5}0.00125 = 3 - 5log_{5}10

Let us consider the RHS:

i.e. 3 - 5log_{5}10 = 3log_{5}5 - 5log_{5}10(∵ log_{a}a = 1)

= log_{5}5^{3} - log_{5}10^{5}

(∵ log_{a}M^{n} = nlog_{a}M)

=

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

= log_{5}0.00125

**Question 46.**Prove the following equations.

**Answer:**

Let us consider the RHS

(∵)

_{=} log_{5}6 - log_{5}2 + 4

= log_{5}6 - log_{5}2 + 4log_{5}5

(∵ log_{a}(M ÷ N) = ( log_{a}M) - (log_{a}N) and log_{a}(M×N) = (log_{a}M ) + (log_{a}N))

= log_{5}1875

Hence LHS = RHS

**Question 1.**

State whether each of the following statements is true or false.

(i) log_{5}125 = 3

(ii)

(iii) log_{4}(6 + 3) = log_{4}6 + log_{4}3

(iv)

(v)

(vi) log_{a}M - N = log_{a}Mlog_{a}N

**Answer:**

(i) True

log_{5}125 = 3

⇒ 5^{3} = 125

(∵ x = log_{a}b is the logarithmic form of the exponential form a^{x} = b)

This is true.

(ii) False

⇒

(∵ x = log_{a}b is the logarithmic form of the exponential form ax = b)

Here

Therefore, this False.

(iii) False

Here its given log_{4}(6 + 3) = log_{4}6 + log_{4}3

Let us consider the RHS i.e.

log_{4}6 + log_{4}3 = log_{4}(6×3) (∵ according to the product rule loga(M×N) = logaM + logaN;

a,M,N are positive numbers,a≠1)

But here LHS is log4 (6 + 3)

Hence it’s False.

(iv) False

Here it’s given

Let us consider the LHS i.e.

(∵ log_{a}M ÷ log_{a}N = log_{a}M - log_{a}N

;a,M,N are positive numbers ,a≠1)

But here the RHS is

Hence both the sides are not equal and therefore it’s False.

(v) True

Here it’s given:

⇒ (∵ x = log_{a}b is the logarithmic form of the exponential form a^{x} = b)

⇒

Hence LHS = RHS

Therefore this is True.

(vi) False

Here it’s given that log_{a} (M - N) = log_{a} M ÷ log_{a}N

Let us consider the RHS

log_{a}M ÷ log_{a}N = log_{a}M - log_{a}N

(∵ according to quotient rule,log_{a}M ÷ log_{a}N = log_{a}M - log_{a}N ;a,M,N are positive numbers,a≠1)

But the LHS is log_{a}(M - N)

Therefore LHS≠RHS

Hence it’s False.

**Question 2.**

Obtain the equivalent logarithmic form of the following.

2^{4} = 16

**Answer:**

Here it’s given that 2^{4} = 16,

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the equation 2^{4} = 16 (a = 2,b = 16 ,x = 4)

⇒ log_{2}16 = 4

**Question 3.**

Obtain the equivalent logarithmic form of the following.

3^{5} = 243

**Answer:**

Here it’s given that 3^{5} = 243

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the equation 3^{5} = 243 (a = 3,b = 343 ,x = 5)

⇒ log_{3}243 = 5

**Question 4.**

Obtain the equivalent logarithmic form of the following.

10^{-1} = 0.1

**Answer:**

Here it’s given that 10 ^{- 1} = 0.1

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the equation 10 ^{- 1} = 0.1 (a = 10,b = 0.1,x = - 1)

⇒

**Question 5.**

Obtain the equivalent logarithmic form of the following.

**Answer:**

Here it’s given that

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the given equation (a = 8,,)

⇒

**Question 6.**

Obtain the equivalent logarithmic form of the following.

**Answer:**

Here it’s given that

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the given equation (a = 25,b = 5,)

⇒

**Question 7.**

Obtain the equivalent logarithmic form of the following.

**Answer:**

Here it’s given that

The given equation is in the form of a^{x} = b.

log_{a}b is the logarithmic form of the exponential form a^{x} = b

In the equation (a = 12,,x = - 2)

⇒

**Question 8.**

Obtain the equivalent exponential form of the following.

log_{6}216 = 3

**Answer:**

Here it’s given that log_{6}216 = 3

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation log_{6}216 = 3 ( a = 6,b = 216 ,x = 3)

⇒ 6^{3} = 216

**Question 9.**

Obtain the equivalent exponential form of the following.

**Answer:**

Here it’s given that

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation ( a = 9,b = 3 ,)

⇒

**Question 10.**

Obtain the equivalent exponential form of the following.

log_{5}1 = 0

**Answer:**

Here it’s given that log_{5}1 = 0

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation log_{5}1 = 0 (a = 5,b = 1,x = 0)

⇒ 5^{0} = 1

**Question 11.**

Obtain the equivalent exponential form of the following.

**Answer:**

Here it’s given that

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation (,b = 9,x = 4)

⇒

**Question 12.**

Obtain the equivalent exponential form of the following.

**Answer:**

Here it’s given that

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation ( a = 64,,)

⇒

**Question 13.**

Obtain the equivalent exponential form of the following.

log_{0.5}8 = - 3

**Answer:**

Here it’s given that log_{0.5}8 = - 3

The given equation is in the form of log_{a}b = x

The exponential form of the logarithmic form log_{a}b is a^{x} = b.

In the given equation log_{0.5}8 = - 3 (a = 0.5,b = 8,x = - 3)

⇒ (0.5) ^{- 3} = 8

**Question 14.**

Find the value of the following

**Answer:**

i.e. log_{3}(3 ^{- 4}) = - 4(log_{3}3)

(∵ nlog_{a}M = log_{a}M^{n})

⇒ - 4(1) = - 4

(log_{a}a = 1)

**Question 15.**

Find the value of the following

log_{7} 343

**Answer:**

log_{7}343 = log_{7}7^{3}

⇒ 3log_{7}7 (∵ nlog_{a}M = log_{a}M^{n})

⇒ 1(∵ log_{a}a = 1)

**Question 16.**

Find the value of the following

log_{6}6^{5}

**Answer:**

log_{6}6^{5}

⇒ 5log_{6}6

(∵ nlog_{a}M = log_{a}M^{n})

= 5(1)

(∵ log_{a}a = 1)

= 5

**Question 17.**

Find the value of the following

**Answer:**

Here we have i.e.

⇒ , here is

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

⇒ 3( - 1) = - 3

**Question 18.**

Find the value of the following

log_{10} 0.0001

**Answer:**

Here we have log100.0001, i.e.

⇒

⇒ - 4log_{10}10 (∵ nlog_{a}M = log_{a}M^{n})

⇒ - 4(1) = - 4 (∵ log_{a}a = 1)

**Question 19.**

Find the value of the following

**Answer:**

Here we have ,

⇒

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

⇒

⇒

⇒ x = 5

Hence the value of is 5.

**Question 20.**

Solve the following equations.

**Answer:**

⇒ i.e

**Question 21.**

Solve the following equations.

**Answer:**

⇒

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

Or

Or

**Question 22.**

Solve the following equations.

log_{3} y = – 2

**Answer:**

log_{3}y = - 2

log_{3}y = - 2

⇒ 3 ^{- 2} = y

⇒ y = 3 ^{- 2}

⇒

i.e.

**Question 23.**

Solve the following equations.

**Answer:**

⇒

⇒

⇒

∴

**Question 24.**

Solve the following equations.

log_{x} 0.001 = – 3

**Answer:**

log_{x}0.001 = - 3

⇒ x ^{- 3} = 0.001

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

⇒

⇒

⇒ x = 10

**Question 25.**

Solve the following equations.

x + 2 log_{27} 9 = 0

**Answer:**

x + 2log_{27}9 = 0

⇒ x = - 2log_{27}9

⇒ x = log_{27}9 ^{- 2}

⇒ x = log_{3}^{3}(3^{2}) ^{- 2}

⇒ x = log_{3}^{3}(3) ^{- 4}

⇒ (3^{3})^{x} = 3 ^{– 4}

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

⇒ 3x = - 4 (compare the exponents)

⇒

**Question 26.**

Simplify the following.

log_{10}3 + log_{10}3

**Answer:**

log_{10}3 + log_{10}3 = log_{10}(3×3) = log_{10}9

(∵ using the product rule,log_{a}(M×N) = (log_{a}M) + (log_{a}N);a,M,N are positive numbers ,a≠1)

**Question 27.**

Simplify the following.

log_{25}35 – log_{25}10

**Answer:**

(using the quotient rule log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N) );a,M,N are positive numbers ,a≠1)

=

**Question 28.**

Simplify the following.

log_{7}21 + log_{7}77 + log_{7}88 – log_{7}121 – log_{7}24

**Answer:**

log_{7}21 + log_{7}77 + log_{7}88 - log_{7}121 - log_{7}24

⇒

(using the product rule and the quotient rule i.e.

log_{a}(M×N) = (log_{a}M) + (log_{a}N) and

log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

⇒

⇒

⇒ log_{7}7^{2}

⇒ 2log_{7}7 = 2

(∵ log_{7}7 = 1 )

**Question 29.**

Simplify the following.

**Answer:**

⇒ log_{8}(16×52) - log_{8}13

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N) and )

⇒

(∵ loga(M ÷ N) = (log_{a}M) - (log_{a}N))

⇒ log_{8}(16×4) = log_{8}64

⇒ 8x = 64 or x = 2

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

**Question 30.**

Simplify the following.

5log_{10}2 + 2log_{10}3 - 6log_{64}4

**Answer:**

5log_{10}2 + 2log_{10}3 - 6log_{64}4

Here log_{64}4 = x

⇒ 64^{x} = 4

⇒ (4^{3})^{x} = 4

⇒ 3x = 1

⇒

∴

∴ 5log_{10}2 + 2log_{10}3 - 6log_{64}4 = log_{10}2^{5} + log_{10}3^{2} - 2

= log_{10}32 + log_{10}9 - 2log_{10}10

= log_{10}32 + log_{10}9 - log_{10}10^{2}

^{=}

**Question 31.**

Simplify the following.

log_{10}8 + log_{10}5 - log_{10}4

**Answer:**

log_{10}8 + log_{10}5 - log_{10}4

⇒ log_{10}(8×5) - log_{10}4

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

⇒

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

⇒ log10(2×5) = log1010 = 1

(∵ log_{a}a = 1)

**Question 32.**

Solve the equation in each of the following.

log_{4}(x + 4) + log_{4}8 = 2

**Answer:**

log_{4}(x + 4) + log_{4}8 = 2

⇒ log_{4}((x + 4)×8) = 2

⇒ log_{4}(8x + 32) = 2

⇒ 8x + 32 = 4^{2}

⇒ 8x + 32 = 16

⇒ 8x = 16 - 32 = - 16

⇒ 8x = - 16

⇒ x = - 2

**Question 33.**

Solve the equation in each of the following.

log_{6}(x + 4) - log_{6}(x - 1) = 2

**Answer:**

log_{6}(x + 4) - log_{6}(x - 1) = 2

⇒

⇒ (x + 4)(x - 1) = 6^{2} = 6×6

⇒ x + 4 = 6

⇒ x = 6 - 4 = 2

**Question 34.**

Solve the equation in each of the following.

**Answer:**

log_{2}x + log_{4}x + log_{8}x =

Here LHS is

⇒ log_{2}x + log_{2}^{2}x + log_{2}^{3}x

⇒

(∵)

⇒

(∵ log_{a}M^{n} = nlog_{a}M)

⇒

⇒

⇒

Now we equate LHS to the RHS i.e.

⇒

⇒ logx2 = 1 or x1 = 2 or x = 2

**Question 35.**

Solve the equation in each of the following.

log_{4}(8log_{2}x) = 2

**Answer:**

log_{4}(8log_{2}x) = 2

⇒ 8log_{2}x = 4^{2}

(∵ a^{x} = b is the exponential form of logarithmic form of log_{a}b)

⇒ log_{2}x^{8} = 16

(∵ log_{a}M^{n} = nlog_{a}M)

⇒ 2^{16} = x^{8}

⇒ (2^{2})^{8} = x^{8}

⇒ x = 2^{2} = 4

**Question 36.**

Solve the equation in each of the following.

log_{10}5 + log_{10}(5x + 1) = log_{10}(x + 5) + 1

**Answer:**

log_{10}5 + log_{10}(5x + 1) = log_{10}(x + 5) + 1

⇒ log_{10}(5(5x + 1)) - log_{10}(x + 5) = 1

⇒

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

⇒

⇒ 25x + 5 = 10(x + 5)

⇒ 25x + 5 = 10x + 50

⇒ 25x - 10x = 50 - 5 = 45

⇒ 15x = 45

⇒ x = 3

**Question 37.**

Solve the equation in each of the following.

4log_{2}x - log_{2}5 = log_{2}125

**Answer:**

4log_{2}x - log_{2}5 = log_{2}125

⇒ log_{2}x^{4} - log_{2}5 = log_{2}125

⇒

⇒

⇒ x^{4} = 5×125 = 5×5^{3} = 5^{4}

⇒ x = 5

**Question 38.**

Solve the equation in each of the following.

log_{3}25 + log_{3}x = 3log_{3}5

**Answer:**

log_{3}25 + log_{3}x = 3log_{3}5

⇒ log_{3}(25×x) = 3log_{3}5

⇒ log_{3}(25x) = log_{3}5^{3}

⇒ 25x = 5^{3} or (5^{2})x = 5^{3}

⇒ x = 5

**Question 39.**

Solve the equation in each of the following.

**Answer:**

⇒

⇒

(∵ log_{a}(M ÷ N) = log_{a}M - log_{a}N)

⇒

(∵ a^{x} = b is the exponential form of logarithmic form log_{a}b)

⇒

⇒

⇒ 5x - 2 = 3(x + 4)

⇒ 5x - 2 = 3x + 12

⇒ 5x - 3x = 12 + 2

⇒ 2x = 14

⇒ x = 7

**Question 40.**

Given log_{a}2 = x, log_{a} 3 = y and log_{a} 5 = z. Find the value in each of the following in terms of x, y and z.

(i) log_{a}15 (ii) log_{a}8 (iii) log_{a}30

(iv) (v) (vi) log_{a}1.5

**Answer:**

(i) log_{a}15 = log_{a}(5×3)

i.e. log_{a}(5×3) = log_{a}5 + log_{a}3

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

= z + y(∵ log_{a}5 = z,log_{a}3 = y)

(ii) log_{a}8 = log_{a}2^{3} = 3log_{a}2 = 3x

(∵ log_{a}2 = x)

(iii) log_{a}30 = log_{a}(5×3×2) = log_{a}(5) + log_{a}(3) + log_{a}(2)

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

= z + y + x

(∵ log_{a}5 = z,log_{a}3 = y,log_{a}2 = x)

= x + y + z

(iv)

⇒ log_{a}(3×3×3) - log_{a}(5×5×5)

⇒ (log_{a}3 + log_{a}3 + log_{a}3) - (log_{a}5 + log_{a}5 + log_{a}5)

⇒ (y + y + y) - (z + z + z) = 3y - 3z = 3(y - z)

(v)

⇒ log_{a}10 - log_{a}3

(∵ log_{a}(M ÷ N) = log_{a}M - log_{a}N)

Here log_{a}10 = log_{a}(5×2)

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

= log_{a}5 + log_{a}2 = z + x (∵ log_{a}5 = z,log_{a}2 = x)

(vi)

⇒

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

= y - x(∵ log_{a}3 = y,log_{a}2 = x)

**Question 41.**

Prove the following equations.

log_{10}1600 = 2 + 4log_{10}2

**Answer:**

log_{10}1600 = 2 + 4log_{10}2 = 2log_{10}10 + 4log_{10}2

Let us consider the RHS:

i.e. 2 + 4log_{10}2 = 2log_{10}10 + 4log_{10}2

(∵ log_{a}a = 1)

= log_{10}10^{2} + log_{10}2^{4}

(∵ log_{a}M^{n} = nlog_{a}M)

= log_{10}100 + log_{10}16

= log_{10}(100×16)

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

= log_{10}1600

Hence LHS = RHS

**Question 42.**

Prove the following equations.

log_{10}12500 = 2 + 3log_{10}5

**Answer:**

log_{10}12500 = 2 + 3log_{10}5 = 2log_{10}10 + 3log_{10}5

Let us consider the RHS:

i.e. 2 + 3log_{10}5 = 2log_{10}10 + 3log_{10}5

= log_{10}10^{2} + log_{10}5^{3}

(∵ log_{a}M^{n} = nlog_{a}M)

= log_{10}(10^{2}×5^{3})

(∵ log_{a}(M×N) = (log_{a}M) + (log_{a}N))

= log_{10}(100×125)

= log_{10}(12500)

Hence LHS = RHS

**Question 43.**

Prove the following equations.

log_{10}2500 = 4 - 2log_{10}2

**Answer:**

log_{10}2500 = 4 - 2log_{10}2

Let us consider the RHS:

i.e. 4 - 2log_{10}2 = 4log_{10}10 - 2log_{10}2

= log_{10}10^{4} - log_{10}2^{2}

(∵ log_{a}M^{n} = nlog_{a}M)

=

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

Hence LHS = RHS

**Question 44.**

Prove the following equations.

log_{10}0.16 = 2log_{10}4 – 2

**Answer:**

Let us consider the RHS:

i.e. 2log_{10}4 - 2 = 2log_{10}4 - 2log_{10}10

= log_{10}4^{2} - log_{10}10^{2}

(∵ log_{a}M^{n} = nlog_{a}M)

=

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

= log_{10}(0.16) = log_{10}0.16

Hence LHS = RHS

**Question 45.**

Prove the following equations.

log_{5}0.00125 = 3 - 5log_{5}10

**Answer:**

log_{5}0.00125 = 3 - 5log_{5}10

Let us consider the RHS:

i.e. 3 - 5log_{5}10 = 3log_{5}5 - 5log_{5}10(∵ log_{a}a = 1)

= log_{5}5^{3} - log_{5}10^{5}

(∵ log_{a}M^{n} = nlog_{a}M)

=

(∵ log_{a}(M ÷ N) = (log_{a}M) - (log_{a}N))

= log_{5}0.00125

**Question 46.**

Prove the following equations.

**Answer:**

Let us consider the RHS

(∵)

_{=} log_{5}6 - log_{5}2 + 4

= log_{5}6 - log_{5}2 + 4log_{5}5

(∵ log_{a}(M ÷ N) = ( log_{a}M) - (log_{a}N) and log_{a}(M×N) = (log_{a}M ) + (log_{a}N))

= log_{5}1875

Hence LHS = RHS

###### Exercise 2.3

**Question 1.**Write each of the following in scientific notation:

92.43

**Answer:**__Scientific Notation:__ A number is written in **scientific notation** when a number between 1 and 10 is multiplied by a power of 10.

Let N = 92.43

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 92.43 = 9.243 × 10^{1}

**Question 2.**Write each of the following in scientific notation:

0.9243

**Answer:**__Scientific Notation:__ A number is written in **scientific notation** when a number between 1 and 10 is multiplied by a power of 10.

Let N = 0.9243

Divide N by 10000 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 0.9243 = 9.243 × 10^{–1}

**Question 3.**Write each of the following in scientific notation:

9243

**Answer:**__Scientific Notation:__ A number is written in **scientific notation** when a number between 1 and 10 is multiplied by a power of 10.

Let N = 9243

Multiply and Divide N by 1000, we get

Thus, scientific notation of 9243 = 9.243 × 10^{3}

**Question 4.**Write each of the following in scientific notation:

924300

**Answer:**__Scientific Notation:__ A number is written in **scientific notation** when a number between 1 and 10 is multiplied by a power of 10.

Let N = 924300

Multiply and Divide N by 10^{5}, we get

Thus, scientific notation of 924300 = 9.243 × 10^{5}

**Question 5.**Write each of the following in scientific notation:

0.009243

**Answer:**Let N = 0.009243

Divide N by 10^{6} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 0.009243 = 9.243 × 10^{–3}

**Question 6.**Write each of the following in scientific notation:

0.09243

**Answer:**__Scientific Notation:__ A number is written in **scientific notation** when a number between 1 and 10 is multiplied by a power of 10.

Let N = 0.09243

Divide N by 10^{5} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 0.09243 = 9.243 × 10^{–2}

**Question 7.**Write the characteristic of each of the following

log 4576

**Answer:**__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Let N = 4576

Multiply and Divide N by 1000, we get

Thus, scientific notation of 4576 = 4.576 × 10^{3}

Consider,

log 4576 = log (4.576 × 10^{3} )

= log 4.576 + log 10^{3}

(since, log (a×b) = log a + log b)

= log 4.576 + 3 (since, log 10^{n} = n)

Thus characteristic of log 4576 is 3

**Question 8.**Write the characteristic of each of the following

log 24.56

**Answer:**__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Let N = 24.56

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 24.56 = 2.456 × 10^{1}

Consider,

log 24.56 = log (2.456 × 10^{1} )

= log 2.456 + log 10^{1}

(since, log (a×b) = log a + log b)

= log 2.456 + 1 (since, log 10^{n} = n)

Thus characteristic of log 24.56 is 1

**Question 9.**Write the characteristic of each of the following

log 0.00257

**Answer:**__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Let N = 0.00257

Divide N by 10^{5} to remove decimal, we get

Multiply and Divide N by 100, we get

Thus, scientific notation 0.00257 = 2.57 × 10^{–3}

Consider,

log 0.00257 = log (2.57 × 10^{–3} )

= log 2.57 + log 10^{–3}

(since, log (a×b) = log a + log b)

= log 2.57 + (–3)

(since, log 10^{n} = n)

Thus characteristic of log 0.00257 is –3

**Question 10.**Write the characteristic of each of the following

log 0.0756

**Answer:**__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Let N = 0.0756

Divide N by 10^{4} to remove decimal, we get

Multiply and Divide N by 100, we get

Thus, scientific notation 0.0756 = 7.56 × 10^{–2}

Consider,

log 0.0756 = log (7.56 × 10^{–2} )

= log 7.56 + log 10^{–2}

(since, log (a×b) = log a + log b)

= log 7.56 + (–2)

(since, log 10^{n} = n)

Thus characteristic of log 0.0756 is –2

**Question 11.**Write the characteristic of each of the following

log 0.2798

**Answer:**__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Let N = 0.2798

Divide N by 10^{4} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.2798 = 2.798 × 10^{–1}

Consider,

log 0.2798 = log (2.798 × 10^{–1} )

= log 2.798 + log 10^{–1}

(since, log (a×b) = log a + log b)

= log 2.798 + (–1)

(since, log 10^{n} = n)

Thus characteristic of log 0.2798 is –1

**Question 12.**Write the characteristic of each of the following

log 6.453

**Answer:**__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Consider,

log 6.453 = lo

g (6.453 × 10^{0} )

= log 6.453 + log 10^{0}

(since, log (a×b) = log a + log b)

= log 6.453 + 0

(since, log 10^{n} = n)

Thus characteristic of log 6.453 is 0

**Question 13.**The mantissa of log 23750 is 0.3756. Find the value of the following.

log 23750

**Answer:**__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 23750

Multiply and Divide N by 10000, we get

Thus, scientific notation of 23750 = 2.3750 × 10^{4}

Consider,

log 23750 = log (2.3750 × 10^{4} )

= log 2.375 + log 10^{4}

(since, log (a×b) = log a + log b)

= log 2.375 + 4

(since, log 10^{n} = n)

Thus characteristic of log 23750 is 4

Thus, Value of log 23750 = 4 + 0.3756 = 4.3756

**Question 14.**The mantissa of log 23750 is 0.3756. Find the value of the following.

log 23.75

**Answer:**__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 23.75

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 23.75 = 2.375 × 10^{1}

Consider,

log 23.75 = log (2.375 × 10^{1} )

= log 2.375 + log 10^{1}

(since, log (a×b) = log a + log b)

= log 2.375 + 1

(since, log 10^{n} = n)

Thus characteristic of log 23.75 is 1

Thus, Value of log 23.75 = 1 + 0.3756 = 1.3756

**Question 15.**The mantissa of log 23750 is 0.3756. Find the value of the following.

log 2.375

**Answer:**__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Consider,

log 2.375 = log (2.375 × 10^{0} )

= log 2.375 + log 10^{0}

(since, log (a×b) = log a + log b)

= log 2.375 + 0

(since, log 10^{n} = n)

Thus characteristic of log 2.375 is 0

Thus, Value of log 2.375 = 0 + 0.3756 = 0.3756

**Question 16.**The mantissa of log 23750 is 0.3756. Find the value of the following.

log 0.2375

**Answer:**__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 0.2375

Divide N by 10000 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.2375 = 2.375 × 10^{–1}

Consider,

log 0.2375 = log (2.375 × 10^{–1} )

= log 2.375 + log 10^{–1}

(since, log (a×b) = log a + log b)

= log 2.375 + (–1)

(since, log 10^{n} = n)

Thus characteristic of log 0.2375 is –1

Thus, Value of log 0.2375 = –1 + 0.3756 = ̅1.3756

**Question 17.**The mantissa of log 23750 is 0.3756. Find the value of the following.

log 23750000

**Answer:**__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 23750000

Multiply and Divide N by 10^{7}, we get

Thus, scientific notation 23750000 = 2.375 × 10^{7}

Consider,

log 23750000 = log (2.375 × 10^{7} )

= log 2.375 + log 10^{7}

(since, log (a×b) = log a + log b)

= log 2.375 + 7

(since, log 10^{n} = n)

Thus characteristic of log 23750000 is 7

Thus, Value of log 23750000 = 7 + 0.3756 = 7.3756

**Question 18.**The mantissa of log 23750 is 0.3756. Find the value of the following.

log 0.00002375

**Answer:**__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 0.00002375

Divide N by 10^{8} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.00002375 = 2.375 × 10^{–5}

Consider,

log 0.00002375 = log (2.375 × 10^{–5} )

= log 2.375 + log 10^{–5}

(since, log (a×b) = log a + log b)

= log 2.375 + (–5)

(since, log 10^{n} = n)

Thus characteristic of log 0.00002375 is –5

Thus, Value of log 0.00002375 = –5 + 0.3756 = ̅5.3756

**Question 19.**Using logarithmic table find the value of the following.

log 23.17

**Answer:**Let N = 23.17

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 23.17 = 2.317 × 10^{1}

Consider,

log 23.17 = log (2.317 × 10^{1} )

= log 2.317 + log 10^{1}

(since, log (a×b) = log a + log b)

= log 2.317 + 1

(since, log 10^{n} = n)

Thus characteristic of log 23.17 is 1

From the table log 2.31 = 0.3636

Mean difference of 7 is 0.0013

Thus, Mantissa of log 23.17 = 0.3636 + 0.0013 = 0.3649

Thus, Value of log 23.17 = 1 + 0.3649 = 1.3649

**Question 20.**Using logarithmic table find the value of the following.

log 9.321

**Answer:**Let N = 9.321

Consider,

log 9.321 = log (9.321 × 10^{0} )

= log 9.321 + log 10^{0}

(since, log (a×b) = log a + log b)

= log 9.321 + 0

(since, log 10^{n} = n)

Thus characteristic of log 9.321 is 0

From the table log 9.32 = 0.9694

Mean difference of 1 is 0

Thus, Mantissa of log 9.321 = 0.9694

Thus, Value of log 9.32 = 0+ 0.9694 = 0.9694

**Question 21.**Using logarithmic table find the value of the following.

log 329.5

**Answer:**Let N = 329.5

Divide N by 10 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 329.5 = 3.295 × 10^{2}

Consider,

log 329.5 = log (3.295 × 10^{2} )

= log 3.295 + log 10^{2}

(since, log (a×b) = log a + log b)

= log 3.295 + 2

(since, log 10^{n} = n)

Thus characteristic of log 329.5 is 2

From the table log 3.29 = 0.5172

Mean difference of 5 is 0.0007

Thus, Mantissa of log 329.5 = 0.5172+0.0007 = 0.5179

Thus, Value of log 329.5 = 2+0.5178 = 2.5179

**Question 22.**Using logarithmic table find the value of the following.

log 0.001364

**Answer:**Let N = 0.001364

Divide N by 10^{6} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.001364 = 1.364 × 10^{–3}

Consider,

log 0.001364 = log (1.364 × 10^{–3} )

= log 1.364 + log 10^{–3}

(since, log (a×b) = log a + log b)

= log 1.364 + (–3)

(since, log 10^{n} = n)

Thus characteristic of log 1.364 is –3

From the table log 1.36 = 0.1335

Mean difference of 4 is 0.0013

Thus, Mantissa of log 0.001364 = 0.1335+0.0013 = 0.1348

Thus, Value of log 0.001364 = –3 + 0.1348 = ̅3.1348

**Question 23.**Using logarithmic table find the value of the following.

log 0.9876

**Answer:**Let N = 0.9876

Divide N by 10^{4} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.9876= 9.876 × 10^{–1}

Consider,

log 0.9876 = log (9.876 × 10^{–1} )

= log 9.876 + log 10^{–1}

(since, log (a×b) = log a + log b)

= log 9.876 + (–1)

(since, log 10^{n} = n)

Thus characteristic of log 0.9876 is –1

From the table log 9.87=0.9943

Mean difference of 6 is 0.0003

Thus, Mantissa of log 0.9876 = 0.9943+0.0003=0.9946

Thus, Value of log 0.9876 = –1+0.9946 = ̅1.9946

**Question 24.**Using logarithmic table find the value of the following.

log 6576

**Answer:**Let N = 6576

Multiply and Divide N by 1000, we get

Thus, scientific notation 6576= 6.576 × 10^{3}

Consider,

log 6576 = log (6.576 × 10^{3} )

= log 6.576 + log 10^{3}

(since, log (a×b) = log a + log b)

= log 6.576 + 3

(since, log 10^{n} = n)

Thus characteristic of log 6576 is 3

From the table log 6.57=0.8176

Mean difference of 6 is 0.0004

Thus, Mantissa of log 6576 = 0.8176 +0.0004=0.8180

Thus, Value of log 6576 = 3+0.8180 = 3.8180

**Question 25.**Using antilogarithmic table find the value of the following.

i. antilog 3.072

ii. antilog 1.759

iii. antilog

iv. antilog

v. antilog 0.2732

vi. antilog

**Answer:**(i) Characteristic is 3

Mantissa is 0.072

From the antilog table antilog 0.072 = 1.180

Now as the characteristic is 3, therefore we will place the decimal after 3+1=4 numbers in 1180

∴ antilog 3.072 = 1180

(ii) Characteristic is 1

Mantissa is 0.759

From the antilog table antilog 0.759 = 5.741

Now as the characteristic is 1, therefore we will place the decimal after 1+1=2 numbers in 5741

∴ antilog 1.759 = 57.41

(iii) Characteristic is ̅1 = –1

Mantissa is 0.3826

From the antilog table antilog 0.382 = 2.410

Mean Value of 6 is 0.003

Thus, antilog 0.3826 = 2.410+0.003 = 2.413

Now as the characteristic is –1, therefore we will move decimal

–1+1=0 places left in 2.413

∴ antilog ̅1.3826 = 0.2413

(iv) Characteristic is ̅3 = –3

Mantissa is 0.6037

From the antilog table antilog 0.603 = 4.009

Mean Value of 7 is 0.006

Thus, antilog 0.6037 = 4.009+0.006 = 4.015

Now as the characteristic is –3,

therefore we will move decimal

–3+1=2 places left in 4.015

∴ antilog ̅3.6037 = 0.004015

(v) Characteristic is 0

Mantissa is 0.2732

From the antilog table antilog 0.273 = 1.875

Mean value 2 is 0.001

Thus, antilog 0.2732 = 1.875+0.001 = 1.876

Now as the characteristic is 0, therefore we will place the decimal after 0+1=1 numbers in 1876

∴ antilog 0.2732 = 1.876

(vi) Characteristic is ̅2 = –2

Mantissa is 0.1798

From the antilog table antilog 0.179 = 1.510

Mean Value of 8 is 0.003

Thus, antilog 0.1798 = 1.510+0.003 = 1.513

Now as the characteristic is –2, therefore we will move decimal

–2+1=1 places left in 1.513

∴ antilog ̅2.1798 = 0.01513

**Question 26.**Evaluate:

816.3 × 37.42

**Answer:**Let x = 816.3 × 37.42

Taking log on both side we get,

⇒ logx = log (816.3 × 37.42)

= log 816.3 + log 37.42 (since, log a× b = log a + log b)

= 2.9118+1.5731

⇒ logx = 4.4849

⇒ x = antilog 4.4849 = 30542

**Question 27.**Evaluate:

816.3 ÷ 37.42

**Answer:**Let x = 816.3 ÷ 37.42

Taking log on both side we get,

⇒ logx = log (816.3 ÷ 37.42)

= log 816.3 – log 37.42 (since, log a ÷ b = log a – log b)

= 2.9118–1.5731

⇒ logx = 1.3387

⇒ x = antilog 1.3387 = 21.812

**Question 28.**Evaluate:

0.000645 × 82.3

**Answer:**Let x = 0.000645 × 82.3

Taking log on both side we get,

⇒ logx = log (0.000645 × 82.3)

= log 0.000645 +log 82.3 (since, log a × b = log a +log b)

= ̅3.1904 + 1.9153

= –3.1904+1.9153

=–1. 2751

⇒ logx = –1.2751 = ̅1 . 2751

⇒ x = antilog ̅1.2751 = 0.05307

**Question 29.**Evaluate:

0.3421 ÷ 0.09782

**Answer:**Let x = 0.3421 ÷ 0.09782

Taking log on both side we get,

⇒ logx = log (0.3421 ÷ 0.09782)

= log 0.3421 – log 0.09782 (since, log a÷b = log a –log b)

= ̅0.4658 – ̅1.00957

= –0.04658 – (–1.00957)

= –0.04658 + 1.00957

=0.54377

⇒ logx = 0.54377

⇒ x = antilog 0.54377= 3.497

**Question 30.**Evaluate:

(50.49)^{5}

**Answer:**Let x = (50.49)^{5}

Taking log on both side

⇒ log x = 5 log (50.49) (∵ log a^{n} = n loga)

= 5 × 1.7032

logx = 8.516

⇒ x = antilog 8.516 = 32810000

**Question 31.**Evaluate:

**Answer:**Let x = ∛561.4

Taking log on both side

(∵ log a^{n} = n loga)

logx = 0.9163

⇒ x = antilog 0.9163 = 8.247

**Question 32.**Evaluate:

**Answer:**Let

Taking log on both side we get,

= log (175.23 × 22.159) – log (1828.56)

(∵ log a÷ b = loga – log b)

= log 175.23 + log 22.159 – log 1828.56

(∵ log a×b = loga + log b)

= 2.2436 + 1.3455 – 3.2621

⇒ log x = 0.327

⇒ x = antilog 0.327 = 2.123

**Question 33.**Evaluate:

**Answer:**Let

Taking log on both side we get,

(∵ log a÷ b = loga – log b)

(∵ log a×b = loga + log b)

(since, log a^{n} = n log a)

= 0.4823 + 0.5725 – 0.833

⇒ log x = 0.2218

⇒ x = antilog 0.2218 = 1.666

**Question 34.**Evaluate:

**Answer:**Let

Taking log on both side

⇒ log x = log ( (76.23)^{3} × ∛1.928 ) – log ((42.75)^{5} × 0.04623)

(∵ log a÷ b = loga – log b)

⇒ log x = log (76.23)^{3} +log ∛1.928 – (log (42.75)^{5} +log 0.04623)

(∵ log a × b = loga + log b)

⇒ log x = log (76.23)^{3} +log ∛1.928 – log (42.75)^{5} –log 0.04623

(since, log a^{n} = n log a)

⇒ log x = 5.6463 + 0.0950 – 8.1545 + 1.3350

⇒ log x = –1.0782 = ̅1.0782

⇒ x = antilog ̅1.0782 = 0.08352

**Question 35.**Evaluate:

**Answer:**Let

Taking log on both side,

(since, log a^{n} = n log a)

(∵ log a÷ b = loga – log b)

(∵ log a × b = loga + log b)

⇒ log x = –0.2255

⇒ x = antilog (–0.2255) = antilog ̅0.2255 = 0.5948

**Question 36.**Evaluate:

log_{9} 63.28

**Answer:**Let log_{9} 63.28 = log_{10}63.28 × log_{9}10

(since, log_{a}M = log_{b}M × log_{a}b)

Then

Taking log on both side

⇒ log x = log 1.8012 – log 0.9542

(∵ log a÷ b = loga – log b)

⇒ log x = 0.2555 – (–0.0203)

= 0.2555+0.0203

= 0.2758

⇒ x = antilog 0.2758 = 1.887

**Question 37.**Evaluate:

log_{3} 7

**Answer:**Let log_{3} 7 = log_{10}7× log_{3}10

(since, log_{a}M = log_{b}M × log_{a}b)

Then

Taking log on both side

⇒ log x = log 0.8450 – log 0.4771

(∵ log a÷ b = loga – log b)

⇒ log x = –0.0731– (–0.3213)

= – 0.0731 + 0.3213

= 0.2482

⇒ x = antilog 0.2482 = 1.771

**Question 1.**

Write each of the following in scientific notation:

92.43

**Answer:**

__Scientific Notation:__ A number is written in **scientific notation** when a number between 1 and 10 is multiplied by a power of 10.

Let N = 92.43

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 92.43 = 9.243 × 10^{1}

**Question 2.**

Write each of the following in scientific notation:

0.9243

**Answer:**

__Scientific Notation:__ A number is written in **scientific notation** when a number between 1 and 10 is multiplied by a power of 10.

Let N = 0.9243

Divide N by 10000 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 0.9243 = 9.243 × 10^{–1}

**Question 3.**

Write each of the following in scientific notation:

9243

**Answer:**

__Scientific Notation:__ A number is written in **scientific notation** when a number between 1 and 10 is multiplied by a power of 10.

Let N = 9243

Multiply and Divide N by 1000, we get

Thus, scientific notation of 9243 = 9.243 × 10^{3}

**Question 4.**

Write each of the following in scientific notation:

924300

**Answer:**

__Scientific Notation:__ A number is written in **scientific notation** when a number between 1 and 10 is multiplied by a power of 10.

Let N = 924300

Multiply and Divide N by 10^{5}, we get

Thus, scientific notation of 924300 = 9.243 × 10^{5}

**Question 5.**

Write each of the following in scientific notation:

0.009243

**Answer:**

Let N = 0.009243

Divide N by 10^{6} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 0.009243 = 9.243 × 10^{–3}

**Question 6.**

Write each of the following in scientific notation:

0.09243

**Answer:**

__Scientific Notation:__ A number is written in **scientific notation** when a number between 1 and 10 is multiplied by a power of 10.

Let N = 0.09243

Divide N by 10^{5} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 0.09243 = 9.243 × 10^{–2}

**Question 7.**

Write the characteristic of each of the following

log 4576

**Answer:**

__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Let N = 4576

Multiply and Divide N by 1000, we get

Thus, scientific notation of 4576 = 4.576 × 10^{3}

Consider,

log 4576 = log (4.576 × 10^{3} )

= log 4.576 + log 10^{3}

(since, log (a×b) = log a + log b)

= log 4.576 + 3 (since, log 10^{n} = n)

Thus characteristic of log 4576 is 3

**Question 8.**

Write the characteristic of each of the following

log 24.56

**Answer:**

__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Let N = 24.56

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 24.56 = 2.456 × 10^{1}

Consider,

log 24.56 = log (2.456 × 10^{1} )

= log 2.456 + log 10^{1}

(since, log (a×b) = log a + log b)

= log 2.456 + 1 (since, log 10^{n} = n)

Thus characteristic of log 24.56 is 1

**Question 9.**

Write the characteristic of each of the following

log 0.00257

**Answer:**

__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Let N = 0.00257

Divide N by 10^{5} to remove decimal, we get

Multiply and Divide N by 100, we get

Thus, scientific notation 0.00257 = 2.57 × 10^{–3}

Consider,

log 0.00257 = log (2.57 × 10^{–3} )

= log 2.57 + log 10^{–3}

(since, log (a×b) = log a + log b)

= log 2.57 + (–3)

(since, log 10^{n} = n)

Thus characteristic of log 0.00257 is –3

**Question 10.**

Write the characteristic of each of the following

log 0.0756

**Answer:**

__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Let N = 0.0756

Divide N by 10^{4} to remove decimal, we get

Multiply and Divide N by 100, we get

Thus, scientific notation 0.0756 = 7.56 × 10^{–2}

Consider,

log 0.0756 = log (7.56 × 10^{–2} )

= log 7.56 + log 10^{–2}

(since, log (a×b) = log a + log b)

= log 7.56 + (–2)

(since, log 10^{n} = n)

Thus characteristic of log 0.0756 is –2

**Question 11.**

Write the characteristic of each of the following

log 0.2798

**Answer:**

__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Let N = 0.2798

Divide N by 10^{4} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.2798 = 2.798 × 10^{–1}

Consider,

log 0.2798 = log (2.798 × 10^{–1} )

= log 2.798 + log 10^{–1}

(since, log (a×b) = log a + log b)

= log 2.798 + (–1)

(since, log 10^{n} = n)

Thus characteristic of log 0.2798 is –1

**Question 12.**

Write the characteristic of each of the following

log 6.453

**Answer:**

__Characteristic:__ In a scientific number, the power of 10 determines the characteristic.

Consider,

log 6.453 = lo

g (6.453 × 10^{0} )

= log 6.453 + log 10^{0}

(since, log (a×b) = log a + log b)

= log 6.453 + 0

(since, log 10^{n} = n)

Thus characteristic of log 6.453 is 0

**Question 13.**

The mantissa of log 23750 is 0.3756. Find the value of the following.

log 23750

**Answer:**

__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 23750

Multiply and Divide N by 10000, we get

Thus, scientific notation of 23750 = 2.3750 × 10^{4}

Consider,

log 23750 = log (2.3750 × 10^{4} )

= log 2.375 + log 10^{4}

(since, log (a×b) = log a + log b)

= log 2.375 + 4

(since, log 10^{n} = n)

Thus characteristic of log 23750 is 4

Thus, Value of log 23750 = 4 + 0.3756 = 4.3756

**Question 14.**

The mantissa of log 23750 is 0.3756. Find the value of the following.

log 23.75

**Answer:**

__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 23.75

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 23.75 = 2.375 × 10^{1}

Consider,

log 23.75 = log (2.375 × 10^{1} )

= log 2.375 + log 10^{1}

(since, log (a×b) = log a + log b)

= log 2.375 + 1

(since, log 10^{n} = n)

Thus characteristic of log 23.75 is 1

Thus, Value of log 23.75 = 1 + 0.3756 = 1.3756

**Question 15.**

The mantissa of log 23750 is 0.3756. Find the value of the following.

log 2.375

**Answer:**

__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Consider,

log 2.375 = log (2.375 × 10^{0} )

= log 2.375 + log 10^{0}

(since, log (a×b) = log a + log b)

= log 2.375 + 0

(since, log 10^{n} = n)

Thus characteristic of log 2.375 is 0

Thus, Value of log 2.375 = 0 + 0.3756 = 0.3756

**Question 16.**

The mantissa of log 23750 is 0.3756. Find the value of the following.

log 0.2375

**Answer:**

__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 0.2375

Divide N by 10000 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.2375 = 2.375 × 10^{–1}

Consider,

log 0.2375 = log (2.375 × 10^{–1} )

= log 2.375 + log 10^{–1}

(since, log (a×b) = log a + log b)

= log 2.375 + (–1)

(since, log 10^{n} = n)

Thus characteristic of log 0.2375 is –1

Thus, Value of log 0.2375 = –1 + 0.3756 = ̅1.3756

**Question 17.**

The mantissa of log 23750 is 0.3756. Find the value of the following.

log 23750000

**Answer:**

__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 23750000

Multiply and Divide N by 10^{7}, we get

Thus, scientific notation 23750000 = 2.375 × 10^{7}

Consider,

log 23750000 = log (2.375 × 10^{7} )

= log 2.375 + log 10^{7}

(since, log (a×b) = log a + log b)

= log 2.375 + 7

(since, log 10^{n} = n)

Thus characteristic of log 23750000 is 7

Thus, Value of log 23750000 = 7 + 0.3756 = 7.3756

**Question 18.**

The mantissa of log 23750 is 0.3756. Find the value of the following.

log 0.00002375

**Answer:**

__Mantissa:__ Every logarithm consist of a fractional part called the mantissa.

Here, The mantissa of log 23750 is 0.3756

Let N = 0.00002375

Divide N by 10^{8} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.00002375 = 2.375 × 10^{–5}

Consider,

log 0.00002375 = log (2.375 × 10^{–5} )

= log 2.375 + log 10^{–5}

(since, log (a×b) = log a + log b)

= log 2.375 + (–5)

(since, log 10^{n} = n)

Thus characteristic of log 0.00002375 is –5

Thus, Value of log 0.00002375 = –5 + 0.3756 = ̅5.3756

**Question 19.**

Using logarithmic table find the value of the following.

log 23.17

**Answer:**

Let N = 23.17

Divide N by 100 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 23.17 = 2.317 × 10^{1}

Consider,

log 23.17 = log (2.317 × 10^{1} )

= log 2.317 + log 10^{1}

(since, log (a×b) = log a + log b)

= log 2.317 + 1

(since, log 10^{n} = n)

Thus characteristic of log 23.17 is 1

From the table log 2.31 = 0.3636

Mean difference of 7 is 0.0013

Thus, Mantissa of log 23.17 = 0.3636 + 0.0013 = 0.3649

Thus, Value of log 23.17 = 1 + 0.3649 = 1.3649

**Question 20.**

Using logarithmic table find the value of the following.

log 9.321

**Answer:**

Let N = 9.321

Consider,

log 9.321 = log (9.321 × 10^{0} )

= log 9.321 + log 10^{0}

(since, log (a×b) = log a + log b)

= log 9.321 + 0

(since, log 10^{n} = n)

Thus characteristic of log 9.321 is 0

From the table log 9.32 = 0.9694

Mean difference of 1 is 0

Thus, Mantissa of log 9.321 = 0.9694

Thus, Value of log 9.32 = 0+ 0.9694 = 0.9694

**Question 21.**

Using logarithmic table find the value of the following.

log 329.5

**Answer:**

Let N = 329.5

Divide N by 10 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 329.5 = 3.295 × 10^{2}

Consider,

log 329.5 = log (3.295 × 10^{2} )

= log 3.295 + log 10^{2}

(since, log (a×b) = log a + log b)

= log 3.295 + 2

(since, log 10^{n} = n)

Thus characteristic of log 329.5 is 2

From the table log 3.29 = 0.5172

Mean difference of 5 is 0.0007

Thus, Mantissa of log 329.5 = 0.5172+0.0007 = 0.5179

Thus, Value of log 329.5 = 2+0.5178 = 2.5179

**Question 22.**

Using logarithmic table find the value of the following.

log 0.001364

**Answer:**

Let N = 0.001364

Divide N by 10^{6} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.001364 = 1.364 × 10^{–3}

Consider,

log 0.001364 = log (1.364 × 10^{–3} )

= log 1.364 + log 10^{–3}

(since, log (a×b) = log a + log b)

= log 1.364 + (–3)

(since, log 10^{n} = n)

Thus characteristic of log 1.364 is –3

From the table log 1.36 = 0.1335

Mean difference of 4 is 0.0013

Thus, Mantissa of log 0.001364 = 0.1335+0.0013 = 0.1348

Thus, Value of log 0.001364 = –3 + 0.1348 = ̅3.1348

**Question 23.**

Using logarithmic table find the value of the following.

log 0.9876

**Answer:**

Let N = 0.9876

Divide N by 10^{4} to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation 0.9876= 9.876 × 10^{–1}

Consider,

log 0.9876 = log (9.876 × 10^{–1} )

= log 9.876 + log 10^{–1}

(since, log (a×b) = log a + log b)

= log 9.876 + (–1)

(since, log 10^{n} = n)

Thus characteristic of log 0.9876 is –1

From the table log 9.87=0.9943

Mean difference of 6 is 0.0003

Thus, Mantissa of log 0.9876 = 0.9943+0.0003=0.9946

Thus, Value of log 0.9876 = –1+0.9946 = ̅1.9946

**Question 24.**

Using logarithmic table find the value of the following.

log 6576

**Answer:**

Let N = 6576

Multiply and Divide N by 1000, we get

Thus, scientific notation 6576= 6.576 × 10^{3}

Consider,

log 6576 = log (6.576 × 10^{3} )

= log 6.576 + log 10^{3}

(since, log (a×b) = log a + log b)

= log 6.576 + 3

(since, log 10^{n} = n)

Thus characteristic of log 6576 is 3

From the table log 6.57=0.8176

Mean difference of 6 is 0.0004

Thus, Mantissa of log 6576 = 0.8176 +0.0004=0.8180

Thus, Value of log 6576 = 3+0.8180 = 3.8180

**Question 25.**

Using antilogarithmic table find the value of the following.

i. antilog 3.072

ii. antilog 1.759

iii. antilog

iv. antilog

v. antilog 0.2732

vi. antilog

**Answer:**

(i) Characteristic is 3

Mantissa is 0.072

From the antilog table antilog 0.072 = 1.180

Now as the characteristic is 3, therefore we will place the decimal after 3+1=4 numbers in 1180

∴ antilog 3.072 = 1180

(ii) Characteristic is 1

Mantissa is 0.759

From the antilog table antilog 0.759 = 5.741

Now as the characteristic is 1, therefore we will place the decimal after 1+1=2 numbers in 5741

∴ antilog 1.759 = 57.41

(iii) Characteristic is ̅1 = –1

Mantissa is 0.3826

From the antilog table antilog 0.382 = 2.410

Mean Value of 6 is 0.003

Thus, antilog 0.3826 = 2.410+0.003 = 2.413

Now as the characteristic is –1, therefore we will move decimal

–1+1=0 places left in 2.413

∴ antilog ̅1.3826 = 0.2413

(iv) Characteristic is ̅3 = –3

Mantissa is 0.6037

From the antilog table antilog 0.603 = 4.009

Mean Value of 7 is 0.006

Thus, antilog 0.6037 = 4.009+0.006 = 4.015

Now as the characteristic is –3,

therefore we will move decimal

–3+1=2 places left in 4.015

∴ antilog ̅3.6037 = 0.004015

(v) Characteristic is 0

Mantissa is 0.2732

From the antilog table antilog 0.273 = 1.875

Mean value 2 is 0.001

Thus, antilog 0.2732 = 1.875+0.001 = 1.876

Now as the characteristic is 0, therefore we will place the decimal after 0+1=1 numbers in 1876

∴ antilog 0.2732 = 1.876

(vi) Characteristic is ̅2 = –2

Mantissa is 0.1798

From the antilog table antilog 0.179 = 1.510

Mean Value of 8 is 0.003

Thus, antilog 0.1798 = 1.510+0.003 = 1.513

Now as the characteristic is –2, therefore we will move decimal

–2+1=1 places left in 1.513

∴ antilog ̅2.1798 = 0.01513

**Question 26.**

Evaluate:

816.3 × 37.42

**Answer:**

Let x = 816.3 × 37.42

Taking log on both side we get,

⇒ logx = log (816.3 × 37.42)

= log 816.3 + log 37.42 (since, log a× b = log a + log b)

= 2.9118+1.5731

⇒ logx = 4.4849

⇒ x = antilog 4.4849 = 30542

**Question 27.**

Evaluate:

816.3 ÷ 37.42

**Answer:**

Let x = 816.3 ÷ 37.42

Taking log on both side we get,

⇒ logx = log (816.3 ÷ 37.42)

= log 816.3 – log 37.42 (since, log a ÷ b = log a – log b)

= 2.9118–1.5731

⇒ logx = 1.3387

⇒ x = antilog 1.3387 = 21.812

**Question 28.**

Evaluate:

0.000645 × 82.3

**Answer:**

Let x = 0.000645 × 82.3

Taking log on both side we get,

⇒ logx = log (0.000645 × 82.3)

= log 0.000645 +log 82.3 (since, log a × b = log a +log b)

= ̅3.1904 + 1.9153

= –3.1904+1.9153

=–1. 2751

⇒ logx = –1.2751 = ̅1 . 2751

⇒ x = antilog ̅1.2751 = 0.05307

**Question 29.**

Evaluate:

0.3421 ÷ 0.09782

**Answer:**

Let x = 0.3421 ÷ 0.09782

Taking log on both side we get,

⇒ logx = log (0.3421 ÷ 0.09782)

= log 0.3421 – log 0.09782 (since, log a÷b = log a –log b)

= ̅0.4658 – ̅1.00957

= –0.04658 – (–1.00957)

= –0.04658 + 1.00957

=0.54377

⇒ logx = 0.54377

⇒ x = antilog 0.54377= 3.497

**Question 30.**

Evaluate:

(50.49)^{5}

**Answer:**

Let x = (50.49)^{5}

Taking log on both side

⇒ log x = 5 log (50.49) (∵ log a^{n} = n loga)

= 5 × 1.7032

logx = 8.516

⇒ x = antilog 8.516 = 32810000

**Question 31.**

Evaluate:

**Answer:**

Let x = ∛561.4

Taking log on both side

(∵ log a^{n} = n loga)

logx = 0.9163

⇒ x = antilog 0.9163 = 8.247

**Question 32.**

Evaluate:

**Answer:**

Let

Taking log on both side we get,

= log (175.23 × 22.159) – log (1828.56)

(∵ log a÷ b = loga – log b)

= log 175.23 + log 22.159 – log 1828.56

(∵ log a×b = loga + log b)

= 2.2436 + 1.3455 – 3.2621

⇒ log x = 0.327

⇒ x = antilog 0.327 = 2.123

**Question 33.**

Evaluate:

**Answer:**

Let

Taking log on both side we get,

(∵ log a÷ b = loga – log b)

(∵ log a×b = loga + log b)

(since, log a^{n} = n log a)

= 0.4823 + 0.5725 – 0.833

⇒ log x = 0.2218

⇒ x = antilog 0.2218 = 1.666

**Question 34.**

Evaluate:

**Answer:**

Let

Taking log on both side

⇒ log x = log ( (76.23)^{3} × ∛1.928 ) – log ((42.75)^{5} × 0.04623)

(∵ log a÷ b = loga – log b)

⇒ log x = log (76.23)^{3} +log ∛1.928 – (log (42.75)^{5} +log 0.04623)

(∵ log a × b = loga + log b)

⇒ log x = log (76.23)^{3} +log ∛1.928 – log (42.75)^{5} –log 0.04623

(since, log a^{n} = n log a)

⇒ log x = 5.6463 + 0.0950 – 8.1545 + 1.3350

⇒ log x = –1.0782 = ̅1.0782

⇒ x = antilog ̅1.0782 = 0.08352

**Question 35.**

Evaluate:

**Answer:**

Let

Taking log on both side,

(since, log a^{n} = n log a)

(∵ log a÷ b = loga – log b)

(∵ log a × b = loga + log b)

⇒ log x = –0.2255

⇒ x = antilog (–0.2255) = antilog ̅0.2255 = 0.5948

**Question 36.**

Evaluate:

log_{9} 63.28

**Answer:**

Let log_{9} 63.28 = log_{10}63.28 × log_{9}10

(since, log_{a}M = log_{b}M × log_{a}b)

Then

Taking log on both side

⇒ log x = log 1.8012 – log 0.9542

(∵ log a÷ b = loga – log b)

⇒ log x = 0.2555 – (–0.0203)

= 0.2555+0.0203

= 0.2758

⇒ x = antilog 0.2758 = 1.887

**Question 37.**

Evaluate:

log_{3} 7

**Answer:**

Let log_{3} 7 = log_{10}7× log_{3}10

(since, log_{a}M = log_{b}M × log_{a}b)

Then

Taking log on both side

⇒ log x = log 0.8450 – log 0.4771

(∵ log a÷ b = loga – log b)

⇒ log x = –0.0731– (–0.3213)

= – 0.0731 + 0.3213

= 0.2482

⇒ x = antilog 0.2482 = 1.771

###### Exercise 2.4

**Question 1.**Convert 45_{10} to base 2

**Answer:**

Thus, 45_{10} = 101101_{2}

**Question 2.**Convert 73_{10} to base 2.

**Answer:**

Thus, 73_{10} = 1001001_{2}

**Question 3.**Convert 1101011_{2} to base 10.

**Answer:**1101011_{2} = 1 × 2^{6} + 1 × 2^{5} + 0 × 2^{4} + 1 × 2^{3} + 0 × 2^{2} +

1 × 2^{1}+ 1 × 2^{0}

= 64 + 32+ 0+8+0+2+1 = 107_{10}

Thus, 1101011_{2}= 107_{10}

**Question 4.**Convert 111_{2} to base 10.

**Answer:**111_{2} = 1 × 2^{2} + 1 × 2^{1} +1 × 2^{0}

= 4 + 2+1 = 7_{10}

Thus, 111_{2}=7_{10}

**Question 5.**Convert 987_{10} to base 5.

**Answer:**

Thus, 987_{10} = 12422_{5}

**Question 6.**Convert 1238_{10} to base 5.

**Answer:**

Thus, 1238_{10} = 14423_{5}

**Question 7.**Convert 10234_{5} to base 10.

**Answer:**10234_{5} = 1 × 5^{4} + 0 × 5^{3} + 2 × 5^{2} + 3 × 5^{1}+ 4 × 5^{0}

= 625 + 0 + 50+15+4 = 694_{10}

Thus, 10234_{5} = 694_{10}

**Question 8.**Convert 211423_{5} to base 10.

**Answer:**211423_{5} = 2 × 5^{5} + 1 × 5^{4} + 1 × 5^{3} + 4 × 5^{2} + 2 × 5^{1}+

3 × 5^{0}

= 6250 + 625+ 125+100+10+3 = 7113_{10}

Thus, 211423_{5} = 7113_{10}

**Question 9.**Convert 98567_{10} to base 8.

**Answer:**

Thus, 98567_{10} = 300407_{8}

**Question 10.**Convert 688_{10} to base 8.

**Answer:**

Thus, 688_{10} = 1260_{8}

**Question 11.**Convert 47156_{8} to base 10.

**Answer:**47156_{8} = 4 × 8^{4} + 7 × 8^{3} + 1 × 8^{2} + 5 × 8^{1} + 6 × 8^{0}

= 16384+3584+64+40+6 = 20078_{10}

Thus, 47156_{8} = 20078_{10}

**Question 12.**Convert 585_{10} to base 2,5 and 8.

**Answer:**

Thus, 585_{10} = 1001001001_{2}

Thus, 585_{10} = 4320_{5}

Thus, 585_{10} = 1111_{8}

**Question 1.**

Convert 45_{10} to base 2

**Answer:**

Thus, 45_{10} = 101101_{2}

**Question 2.**

Convert 73_{10} to base 2.

**Answer:**

Thus, 73_{10} = 1001001_{2}

**Question 3.**

Convert 1101011_{2} to base 10.

**Answer:**

1101011_{2} = 1 × 2^{6} + 1 × 2^{5} + 0 × 2^{4} + 1 × 2^{3} + 0 × 2^{2} +

1 × 2^{1}+ 1 × 2^{0}

= 64 + 32+ 0+8+0+2+1 = 107_{10}

Thus, 1101011_{2}= 107_{10}

**Question 4.**

Convert 111_{2} to base 10.

**Answer:**

111_{2} = 1 × 2^{2} + 1 × 2^{1} +1 × 2^{0}

= 4 + 2+1 = 7_{10}

Thus, 111_{2}=7_{10}

**Question 5.**

Convert 987_{10} to base 5.

**Answer:**

Thus, 987_{10} = 12422_{5}

**Question 6.**

Convert 1238_{10} to base 5.

**Answer:**

Thus, 1238_{10} = 14423_{5}

**Question 7.**

Convert 10234_{5} to base 10.

**Answer:**

10234_{5} = 1 × 5^{4} + 0 × 5^{3} + 2 × 5^{2} + 3 × 5^{1}+ 4 × 5^{0}

= 625 + 0 + 50+15+4 = 694_{10}

Thus, 10234_{5} = 694_{10}

**Question 8.**

Convert 211423_{5} to base 10.

**Answer:**

211423_{5} = 2 × 5^{5} + 1 × 5^{4} + 1 × 5^{3} + 4 × 5^{2} + 2 × 5^{1}+

3 × 5^{0}

= 6250 + 625+ 125+100+10+3 = 7113_{10}

Thus, 211423_{5} = 7113_{10}

**Question 9.**

Convert 98567_{10} to base 8.

**Answer:**

Thus, 98567_{10} = 300407_{8}

**Question 10.**

Convert 688_{10} to base 8.

**Answer:**

Thus, 688_{10} = 1260_{8}

**Question 11.**

Convert 47156_{8} to base 10.

**Answer:**

47156_{8} = 4 × 8^{4} + 7 × 8^{3} + 1 × 8^{2} + 5 × 8^{1} + 6 × 8^{0}

= 16384+3584+64+40+6 = 20078_{10}

Thus, 47156_{8} = 20078_{10}

**Question 12.**

Convert 585_{10} to base 2,5 and 8.

**Answer:**

Thus, 585_{10} = 1001001001_{2}

Thus, 585_{10} = 4320_{5}

Thus, 585_{10} = 1111_{8}

###### Exercise 2.5

**Question 1.**The scientific notation of 923.4 is

A. 9.234 × 10^{–2}

B. 9.234 × 10^{2}

C. 9.234 × 10^{3}

D. 9.234 × 10^{–3}

**Answer:**Let N = 923.4

Divide N by 10 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 923.4 = 9.234 × 10^{2}

Correct answer is (B)

**Question 2.**The scientific notation of 0.00036 is

A. 3.6 × 10^{–3}

B. 3.6 × 10^{3}

C. 3.6 × 10^{–4}

D. 3.6 × 10^{4}

**Answer:**Let N = 0.00036

Divide N by 10^{5} to remove decimal, we get

Multiply and Divide N by 10, we get

Thus, scientific notation of 0.00036= 3.6 × 10^{–4}

Correct answer is (C)

**Question 3.**The decimal form of 2.57 x 10^{3}is

A. 257

B. 2570

C. 25700

D. 257000

**Answer:**2.57 x 10^{3} = 2.57 × 1000 = 2570

Correct answer is (B)

**Question 4.**The decimal form of 3.506 × 10^{–2} is

A. 0.03506

B. 0.003506

C. 35.06

D. 350.6

**Answer:**

Correct answer is (A)

**Question 5.**The logarithmic form of 5^{2} = 25 is

A. log_{5}2 = 25

B. log_{2}5 = 25

C. log_{5}25 =2

D. log_{25}5 = 2

**Answer:**We know that x = log_{a}b is the logarithmic form of the exponential form b = a^{x}

Thus, here exponential form 5^{2} = 25 is given

Where b = 25, a =5, x =2

Thus, its logarithmic form is 2 = log_{5} 25

Hence, correct answer is (C)

**Question 6.**The exponential form of log_{2}16 = 4 is

A. 2^{4} = 16

B. 4^{2} = 16

C. 2^{16} = 4

D. 4^{16} = 2

**Answer:**We know that x = log_{a}b is the logarithmic form of the exponential form b = a^{x}

Thus, here logarithmic form log_{2}16 = 4 is given

Where b = 16, a =2, x =4

Thus, its logarithmic form is 2^{4} = 16

Hence, correct answer is (A)

**Question 7.**The value of is

A. – 2

B. 1

C. 2

D. – 1

**Answer:**Ans. Let

Thus, its exponential form is

On equating power of the base we get,

⇒ x = –1

Thus, correct answer is (D)

**Question 8.**The value of is

A. 2

B.

C.

D. 1

**Answer:**Let x = log_{49}7

Thus, its exponential form is

⇒ 49^{x} = 7

⇒ (7^{2})^{x} = 7

⇒ 7^{2x} = 7

On equating power of the base 7 we get,

⇒ 2x = 1

Thus, correct answer is (B)

**Question 9.**The value of is

A. – 2

B. 0

C.

D. 2

**Answer:**Let

Thus, its exponential form is

On equating power of the base we get,

⇒ x = –2

Thus, correct answer is (A)

**Question 10.**log_{10}8 + log_{10}5– log_{10}4 =

A. log_{10}9

B. log_{10}36

C. 1

D. – 1

**Answer:**Consider, log_{10}8 + log_{10}5– log_{10}4 = log_{10}(8× 5) – log_{10}4

(since, log_{a}M + log_{a}N = log_{a}(M× N) )

⇒ log_{10}8 + log_{10}5– log_{10}4 = log_{10}(40) – log_{10}4

= log_{10}(40 ÷ 4)

(since, log_{a}M – log_{a}N = log_{a}(M÷N) )

⇒ log_{10}8 + log_{10}5– log_{10}4 = log_{10}(10) = 1 (since, log_{a}a = 1)

Thus, correct answer is (C)

**Question 1.**

The scientific notation of 923.4 is

A. 9.234 × 10^{–2}

B. 9.234 × 10^{2}

C. 9.234 × 10^{3}

D. 9.234 × 10^{–3}

**Answer:**

Let N = 923.4

Divide N by 10 to remove decimal, we get

Multiply and Divide N by 1000, we get

Thus, scientific notation of 923.4 = 9.234 × 10^{2}

Correct answer is (B)

**Question 2.**

The scientific notation of 0.00036 is

A. 3.6 × 10^{–3}

B. 3.6 × 10^{3}

C. 3.6 × 10^{–4}

D. 3.6 × 10^{4}

**Answer:**

Let N = 0.00036

Divide N by 10^{5} to remove decimal, we get

Multiply and Divide N by 10, we get

Thus, scientific notation of 0.00036= 3.6 × 10^{–4}

Correct answer is (C)

**Question 3.**

The decimal form of 2.57 x 10^{3}is

A. 257

B. 2570

C. 25700

D. 257000

**Answer:**

2.57 x 10^{3} = 2.57 × 1000 = 2570

Correct answer is (B)

**Question 4.**

The decimal form of 3.506 × 10^{–2} is

A. 0.03506

B. 0.003506

C. 35.06

D. 350.6

**Answer:**

Correct answer is (A)

**Question 5.**

The logarithmic form of 5^{2} = 25 is

A. log_{5}2 = 25

B. log_{2}5 = 25

C. log_{5}25 =2

D. log_{25}5 = 2

**Answer:**

We know that x = log_{a}b is the logarithmic form of the exponential form b = a^{x}

Thus, here exponential form 5^{2} = 25 is given

Where b = 25, a =5, x =2

Thus, its logarithmic form is 2 = log_{5} 25

Hence, correct answer is (C)

**Question 6.**

The exponential form of log_{2}16 = 4 is

A. 2^{4} = 16

B. 4^{2} = 16

C. 2^{16} = 4

D. 4^{16} = 2

**Answer:**

We know that x = log_{a}b is the logarithmic form of the exponential form b = a^{x}

Thus, here logarithmic form log_{2}16 = 4 is given

Where b = 16, a =2, x =4

Thus, its logarithmic form is 2^{4} = 16

Hence, correct answer is (A)

**Question 7.**

The value of is

A. – 2

B. 1

C. 2

D. – 1

**Answer:**

Ans. Let

Thus, its exponential form is

On equating power of the base we get,

⇒ x = –1

Thus, correct answer is (D)

**Question 8.**

The value of is

A. 2

B.

C.

D. 1

**Answer:**

Let x = log_{49}7

Thus, its exponential form is

⇒ 49^{x} = 7

⇒ (7^{2})^{x} = 7

⇒ 7^{2x} = 7

On equating power of the base 7 we get,

⇒ 2x = 1

Thus, correct answer is (B)

**Question 9.**

The value of is

A. – 2

B. 0

C.

D. 2

**Answer:**

Let

Thus, its exponential form is

On equating power of the base we get,

⇒ x = –2

Thus, correct answer is (A)

**Question 10.**

log_{10}8 + log_{10}5– log_{10}4 =

A. log_{10}9

B. log_{10}36

C. 1

D. – 1

**Answer:**

Consider, log_{10}8 + log_{10}5– log_{10}4 = log_{10}(8× 5) – log_{10}4

(since, log_{a}M + log_{a}N = log_{a}(M× N) )

⇒ log_{10}8 + log_{10}5– log_{10}4 = log_{10}(40) – log_{10}4

= log_{10}(40 ÷ 4)

(since, log_{a}M – log_{a}N = log_{a}(M÷N) )

⇒ log_{10}8 + log_{10}5– log_{10}4 = log_{10}(10) = 1 (since, log_{a}a = 1)

Thus, correct answer is (C)