Scientific Notations Of Real Numbers And Logarithms Class 9th Mathematics Term 3 Tamilnadu Board Solution

Class 9th Mathematics Term 3 Tamilnadu Board Solution
Exercise 2.1
  1. 749300000000 Represent the following numbers in the scientific notation.…
  2. 13000000 Represent the following numbers in the scientific notation.…
  3. 105003 Represent the following numbers in the scientific notation.…
  4. 543600000000000 Represent the following numbers in the scientific notation.…
  5. 0.0096 Represent the following numbers in the scientific notation.…
  6. 0.0000013307 Represent the following numbers in the scientific notation.…
  7. 0.0000000022 Represent the following numbers in the scientific notation.…
  8. 0.0000000000009 Represent the following numbers in the scientific notation.…
  9. 3.25 × 10-6 Write the following numbers in decimal form.
  10. 4.134 × 10-4 Write the following numbers in decimal form.
  11. 4.134 × 10^4 Write the following numbers in decimal form.
  12. 1.86 × 10^7 Write the following numbers in decimal form.
  13. 9.87 × 10^9 Write the following numbers in decimal form.
  14. 1.432 × 10-9 Write the following numbers in decimal form.
  15. (1000)^2 × (20)^6 Represent the following numbers in scientific notation.…
  16. (1500)^3 (0.0001)^2 Represent the following numbers in scientific notation.…
  17. (16000)^3 ÷ (200)^4 Represent the following numbers in scientific notation.…
  18. (0.003)^7 (0.0002)^5 ÷ (0.001)^3 Represent the following numbers in scientific…
  19. (11000)^3 (0.003)^2 ÷ (30000) Represent the following numbers in scientific…
Exercise 2.2
  1. State whether each of the following statements is true or false. (i) log5125 = 3…
  2. 2^4 = 16 Obtain the equivalent logarithmic form of the following.…
  3. 3^5 = 243 Obtain the equivalent logarithmic form of the following.…
  4. 10-1 = 0.1 Obtain the equivalent logarithmic form of the following.…
  5. 8^- 2/3 = 1/4 Obtain the equivalent logarithmic form of the following.…
  6. 25^1/2 = 5 Obtain the equivalent logarithmic form of the following.…
  7. 12^-2 = 1/144 Obtain the equivalent logarithmic form of the following.…
  8. log6216 = 3 Obtain the equivalent exponential form of the following.…
  9. log_93 = 1/2 Obtain the equivalent exponential form of the following.…
  10. log51 = 0 Obtain the equivalent exponential form of the following.…
  11. log_ root 3 9 = 4 Obtain the equivalent exponential form of the following.…
  12. log_64 (1/8) = - 1/2 Obtain the equivalent exponential form of the following.…
  13. log0.58 = - 3 Obtain the equivalent exponential form of the following.…
  14. log_3 (1/81) Find the value of the following
  15. log7 343 Find the value of the following
  16. log66^5 Find the value of the following
  17. log_ 1/2 8 Find the value of the following
  18. log10 0.0001 Find the value of the following
  19. log_ root 3 9 root 3 Find the value of the following
  20. log_2x = 1/2 Solve the following equations.
  21. log_ 1/2 x = 3 Solve the following equations.
  22. log3 y = - 2 Solve the following equations.
  23. log_x125 root 5 = 7 Solve the following equations.
  24. logx 0.001 = - 3 Solve the following equations.
  25. x + 2 log27 9 = 0 Solve the following equations.
  26. log103 + log103 Simplify the following.
  27. log2535 - log2510 Simplify the following.
  28. log721 + log777 + log788 - log7121 - log724 Simplify the following.…
  29. log_816+log_852 - 1/log_138 Simplify the following.
  30. 5log102 + 2log103 - 6log644 Simplify the following.
  31. log108 + log105 - log104 Simplify the following.
  32. log4(x + 4) + log48 = 2 Solve the equation in each of the following.…
  33. log6(x + 4) - log6(x - 1) = 2 Solve the equation in each of the following.…
  34. log_2x+log_4x+log_8x = 11/6 Solve the equation in each of the following.…
  35. log4(8log2x) = 2 Solve the equation in each of the following.
  36. log105 + log10(5x + 1) = log10(x + 5) + 1 Solve the equation in each of the…
  37. 4log2x - log25 = log2125 Solve the equation in each of the following.…
  38. log325 + log3x = 3log35 Solve the equation in each of the following.…
  39. log_3 (root 5x-2) - 1/2 = log_3 (root x+4) Solve the equation in each of the…
  40. Given loga2 = x, loga 3 = y and loga 5 = z. Find the value in each of the…
  41. log101600 = 2 + 4log102 Prove the following equations.
  42. log1012500 = 2 + 3log105 Prove the following equations.
  43. log102500 = 4 - 2log102 Prove the following equations.
  44. log100.16 = 2log104 - 2 Prove the following equations.
  45. log50.00125 = 3 - 5log510 Prove the following equations.
  46. log_51875 = 1/2 log_536 - 1/3 log_58+20log_322 Prove the following equations.…
Exercise 2.3
  1. 92.43 Write each of the following in scientific notation:
  2. 0.9243 Write each of the following in scientific notation:
  3. 9243 Write each of the following in scientific notation:
  4. 924300 Write each of the following in scientific notation:
  5. 0.009243 Write each of the following in scientific notation:
  6. 0.09243 Write each of the following in scientific notation:
  7. log 4576 Write the characteristic of each of the following
  8. log 24.56 Write the characteristic of each of the following
  9. log 0.00257 Write the characteristic of each of the following
  10. log 0.0756 Write the characteristic of each of the following
  11. log 0.2798 Write the characteristic of each of the following
  12. log 6.453 Write the characteristic of each of the following
  13. log 23750 The mantissa of log 23750 is 0.3756. Find the value of the following.…
  14. log 23.75 The mantissa of log 23750 is 0.3756. Find the value of the following.…
  15. log 2.375 The mantissa of log 23750 is 0.3756. Find the value of the following.…
  16. log 0.2375 The mantissa of log 23750 is 0.3756. Find the value of the…
  17. log 23750000 The mantissa of log 23750 is 0.3756. Find the value of the…
  18. log 0.00002375 The mantissa of log 23750 is 0.3756. Find the value of the…
  19. log 23.17 Using logarithmic table find the value of the following.…
  20. log 9.321 Using logarithmic table find the value of the following.…
  21. log 329.5 Using logarithmic table find the value of the following.…
  22. log 0.001364 Using logarithmic table find the value of the following.…
  23. log 0.9876 Using logarithmic table find the value of the following.…
  24. log 6576 Using logarithmic table find the value of the following.…
  25. Using antilogarithmic table find the value of the following. i. antilog 3.072…
  26. 816.3 × 37.42 Evaluate:
  27. 816.3 ÷ 37.42 Evaluate:
  28. 0.000645 × 82.3 Evaluate:
  29. 0.3421 ÷ 0.09782 Evaluate:
  30. (50.49)^5 Evaluate:
  31. cube root 561.4 Evaluate:
  32. 175.23 x 22.159/1828.56 Evaluate:
  33. cube root 28 x root [5]729/root 46.35 Evaluate:
  34. (76.25)^3 x cube root 1.928/(42.75)^5 x 0.04623 Evaluate:
  35. cube root 0.7214 x 20.37/69.8 Evaluate:
  36. log9 63.28 Evaluate:
  37. log3 7 Evaluate:
Exercise 2.4
  1. Convert 4510 to base 2
  2. Convert 7310 to base 2.
  3. Convert 11010112 to base 10.
  4. Convert 1112 to base 10.
  5. Convert 98710 to base 5.
  6. Convert 123810 to base 5.
  7. Convert 102345 to base 10.
  8. Convert 2114235 to base 10.
  9. Convert 9856710 to base 8.
  10. Convert 68810 to base 8.
  11. Convert 471568 to base 10.
  12. Convert 58510 to base 2,5 and 8.
Exercise 2.5
  1. The scientific notation of 923.4 isA. 9.234 × 10-2 B. 9.234 × 10^2 C. 9.234 ×…
  2. The scientific notation of 0.00036 isA. 3.6 × 10-3 B. 3.6 × 10^3 C. 3.6 × 10-4…
  3. The decimal form of 2.57 x 10^3 isA. 257 B. 2570 C. 25700 D. 257000…
  4. The decimal form of 3.506 × 10-2 isA. 0.03506 B. 0.003506 C. 35.06 D. 350.6…
  5. The logarithmic form of 5^2 = 25 isA. log52 = 25 B. log25 = 25 C. log525 =2 D.…
  6. The exponential form of log216 = 4 isA. 2^4 = 16 B. 4^2 = 16 C. 2^16 = 4 D. 4^16…
  7. The value of log_ 3/4 (4/3) isA. - 2 B. 1 C. 2 D. - 1
  8. The value of log_497 isA. 2 B. 1/2 C. 1/7 D. 1
  9. The value of log_ 1/2 4 isA. - 2 B. 0 C. 1/2 D. 2
  10. 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 7.49300000000×1011 = 7.493×1011.



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 1.3000000×107 = 1.3×107



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 1.05003×105



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 5.436×1014.



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 × 104


Answer:

In decimal form, the given expression is written as:


4.134 × 104


= 41.34 × 103


= 413.4 × 102


= 4134 × 101


= 41340


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



Question 12.

Write the following numbers in decimal form.

1.86 × 107


Answer:

The given number is 1.86×107.


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 × 109


Answer:

The given number is 9.87×109


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 = (1.0×103) and 20 = (2.0×101)6


∴(1000)2×(20)6 = (1.0×103)2×(2.0×101)6


= (1.0)2×(103)2×(2.0)6×(101)6


= 1×106×64×106


= 64×1012


= 6.4×101×1012


= 6.4×1013


∴ (1000)2 x (20)6 in scientific notation is 6.4×1013



Question 16.

Represent the following numbers in scientific notation.

(1500)3(0.0001)2


Answer:

In scientific notation,


1500 = (1.5×103) and 0.0001 = (1.0×10 - 4)


∴(1500)3×(0.0001)2 = (1.5×103)3×(1.0×10 - 4)2


= (1.5)3×(103)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 3.375×101



Question 17.

Represent the following numbers in scientific notation.

(16000)3 ÷ (200)4


Answer:

In scientific notation,


16000 = (1.6×103) and 200 = (2.0×102)


∴ (16000)3 (200)4 = (1.6×104)3 ÷ (2.0×102)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)


⇒ 




1.331×106×3×10 - 5


= 3.993×101


∴ (11000)3 (0.003)2 ÷ (3000) in scientific notation is 3.993×101




Exercise 2.2
Question 1.

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

(i) log5125 = 3

(ii) 

(iii) log4(6 + 3) = log46 + log43

(iv) 

(v) 

(vi) logaM - N = logaMlogaN


Answer:

(i) True


log5125 = 3


⇒ 53 = 125


(∵ x = logab is the logarithmic form of the exponential form ax = b)


This is true.


(ii) False



⇒ 


(∵ x = logab is the logarithmic form of the exponential form ax = b)


Here


Therefore, this False.


(iii) False


Here its given log4(6 + 3) = log46 + log43


Let us consider the RHS i.e.


log46 + log43 = log4(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.



(∵ logaM ÷ logaN = logaM - logaN


;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 = logab is the logarithmic form of the exponential form ax = b)


⇒ 


Hence LHS = RHS


Therefore this is True.


(vi) False


Here it’s given that loga (M - N) = loga M ÷ logaN


Let us consider the RHS


logaM ÷ logaN = logaM - logaN


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


But the LHS is loga(M - N)


Therefore LHS≠RHS


Hence it’s False.



Question 2.

Obtain the equivalent logarithmic form of the following.

24 = 16


Answer:

Here it’s given that 24 = 16,


The given equation is in the form of ax = b.


logab is the logarithmic form of the exponential form ax = b


In the equation 24 = 16 (a = 2,b = 16 ,x = 4)


⇒ log216 = 4



Question 3.

Obtain the equivalent logarithmic form of the following.

35 = 243


Answer:

Here it’s given that 35 = 243


The given equation is in the form of ax = b.


logab is the logarithmic form of the exponential form ax = b


In the equation 35 = 243 (a = 3,b = 343 ,x = 5)


⇒ log3243 = 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 ax = b.


logab is the logarithmic form of the exponential form ax = 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 ax = b.


logab is the logarithmic form of the exponential form ax = 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 ax = b.


logab is the logarithmic form of the exponential form ax = 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 ax = b.


logab is the logarithmic form of the exponential form ax = b


In the equation  (a = 12,,x = - 2)


⇒ 



Question 8.

Obtain the equivalent exponential form of the following.

log6216 = 3


Answer:

Here it’s given that log6216 = 3


The given equation is in the form of logab = x


The exponential form of the logarithmic form logab is ax = b.


In the given equation log6216 = 3 ( a = 6,b = 216 ,x = 3)


⇒ 63 = 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 logab = x


The exponential form of the logarithmic form logab is ax = b.


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


⇒ 



Question 10.

Obtain the equivalent exponential form of the following.

log51 = 0


Answer:

Here it’s given that log51 = 0


The given equation is in the form of logab = x


The exponential form of the logarithmic form logab is ax = b.


In the given equation log51 = 0 (a = 5,b = 1,x = 0)


⇒ 50 = 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 logab = x


The exponential form of the logarithmic form logab is ax = 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 logab = x


The exponential form of the logarithmic form logab is ax = b.


In the given equation  ( a = 64,,)


⇒ 



Question 13.

Obtain the equivalent exponential form of the following.

log0.58 = - 3


Answer:

Here it’s given that log0.58 = - 3


The given equation is in the form of logab = x


The exponential form of the logarithmic form logab is ax = b.


In the given equation log0.58 = - 3 (a = 0.5,b = 8,x = - 3)


⇒ (0.5) - 3 = 8



Question 14.

Find the value of the following



Answer:


i.e. log3(3 - 4) = - 4(log33)


(∵ nlogaM = logaMn)


⇒ - 4(1) = - 4


(logaa = 1)



Question 15.

Find the value of the following

log7 343


Answer:

log7343 = log773


⇒ 3log77 (∵ nlogaM = logaMn)


⇒ 1(∵ logaa = 1)



Question 16.

Find the value of the following

log665


Answer:

log665


⇒ 5log66


(∵ nlogaM = logaMn)


= 5(1)


(∵ logaa = 1)


= 5



Question 17.

Find the value of the following



Answer:

Here we have  i.e. 


⇒ , here  is


(∵ ax = b is the exponential form of logarithmic form of logab)


⇒ 3( - 1) = - 3



Question 18.

Find the value of the following

log10 0.0001


Answer:

Here we have log100.0001, i.e.



⇒ 


⇒ - 4log1010 (∵ nlogaM = logaMn)


⇒ - 4(1) = - 4 (∵ logaa = 1)



Question 19.

Find the value of the following



Answer:

Here we have ,


⇒ 


(∵ ax = b is the exponential form of logarithmic form of logab)


⇒ 


⇒ 


⇒ x = 5


Hence the value of  is 5.



Question 20.

Solve the following equations.



Answer:


⇒  i.e 



Question 21.

Solve the following equations.



Answer:


⇒ 


(∵ ax = b is the exponential form of logarithmic form of logab)


Or 


Or 



Question 22.

Solve the following equations.

log3 y = – 2


Answer:

log3y = - 2


log3y = - 2


⇒ 3 - 2 = y


⇒ y = 3 - 2


⇒ 


i.e.



Question 23.

Solve the following equations.



Answer:


⇒ 


⇒ 


⇒ 




Question 24.

Solve the following equations.

logx 0.001 = – 3


Answer:

logx0.001 = - 3


⇒ x - 3 = 0.001


(∵ ax = b is the exponential form of logarithmic form of logab)


⇒ 


⇒ 


⇒ x = 10



Question 25.

Solve the following equations.

x + 2 log27 9 = 0


Answer:

x + 2log279 = 0


⇒ x = - 2log279


⇒ x = log27- 2


⇒ x = log33(32- 2


⇒ x = log33(3) - 4


⇒ (33)x = 3 – 4


(∵ ax = b is the exponential form of logarithmic form of logab)


⇒ 3x = - 4 (compare the exponents)


⇒ 



Question 26.

Simplify the following.

log103 + log103


Answer:

log103 + log103 = log10(3×3) = log109


(∵ using the product rule,loga(M×N) = (logaM) + (logaN);a,M,N are positive numbers ,a≠1)



Question 27.

Simplify the following.

log2535 – log2510


Answer:


(using the quotient rule loga(M ÷ N) = (logaM) - (logaN) );a,M,N are positive numbers ,a≠1)





Question 28.

Simplify the following.

log721 + log777 + log788 – log7121 – log724


Answer:

log721 + log777 + log788 - log7121 - log724


⇒ 


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


loga(M×N) = (logaM) + (logaN) and


loga(M ÷ N) = (logaM) - (logaN))


⇒ 


⇒ 


⇒ log772


⇒ 2log77 = 2


(∵ log77 = 1 )



Question 29.

Simplify the following.



Answer:


⇒ log8(16×52) - log813


(∵ loga(M×N) = (logaM) + (logaN) and )


⇒ 


(∵ loga(M ÷ N) = (logaM) - (logaN))


⇒ log8(16×4) = log864


⇒ 8x = 64 or x = 2


(∵ ax = b is the exponential form of logarithmic form of logab)



Question 30.

Simplify the following.

5log102 + 2log103 - 6log644


Answer:

5log102 + 2log103 - 6log644


Here log644 = x


⇒ 64x = 4


⇒ (43)x = 4


⇒ 3x = 1


⇒ 


∴ 


∴ 5log102 + 2log103 - 6log644 = log1025 + log1032 - 2


= log1032 + log109 - 2log1010


= log1032 + log109 - log10102


=



Question 31.

Simplify the following.

log108 + log105 - log104


Answer:

log108 + log105 - log104


⇒ log10(8×5) - log104


(∵ loga(M×N) = (logaM) + (logaN))


⇒ 


(∵ loga(M ÷ N) = (logaM) - (logaN))


⇒ log10(2×5) = log1010 = 1


(∵ logaa = 1)



Question 32.

Solve the equation in each of the following.

log4(x + 4) + log48 = 2


Answer:

log4(x + 4) + log48 = 2


⇒ log4((x + 4)×8) = 2


⇒ log4(8x + 32) = 2


⇒ 8x + 32 = 42


⇒ 8x + 32 = 16


⇒ 8x = 16 - 32 = - 16


⇒ 8x = - 16


⇒ x = - 2



Question 33.

Solve the equation in each of the following.

log6(x + 4) - log6(x - 1) = 2


Answer:

log6(x + 4) - log6(x - 1) = 2


⇒ 


⇒ (x + 4)(x - 1) = 62 = 6×6


⇒ x + 4 = 6


⇒ x = 6 - 4 = 2



Question 34.

Solve the equation in each of the following.



Answer:

log2x + log4x + log8x =



Here LHS is 


⇒ log2x + log22x + log23x


⇒ 


(∵)


⇒ 


(∵ logaMn = nlogaM)


⇒ 


⇒ 


⇒ 


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.

log4(8log2x) = 2


Answer:

log4(8log2x) = 2


⇒ 8log2x = 42


(∵ ax = b is the exponential form of logarithmic form of logab)


⇒ log2x8 = 16


(∵ logaMn = nlogaM)


⇒ 216 = x8


⇒ (22)8 = x8


⇒ x = 22 = 4



Question 36.

Solve the equation in each of the following.

log105 + log10(5x + 1) = log10(x + 5) + 1


Answer:

log105 + log10(5x + 1) = log10(x + 5) + 1


⇒ log10(5(5x + 1)) - log10(x + 5) = 1


⇒ 


(∵ loga(M ÷ N) = (logaM) - (logaN))


⇒ 


⇒ 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.

4log2x - log25 = log2125


Answer:

4log2x - log25 = log2125


⇒ log2x4 - log25 = log2125


⇒ 


⇒ 


⇒ x4 = 5×125 = 5×53 = 54


⇒ x = 5



Question 38.

Solve the equation in each of the following.

log325 + log3x = 3log35


Answer:

log325 + log3x = 3log35


⇒ log3(25×x) = 3log35


⇒ log3(25x) = log353


⇒ 25x = 53 or (52)x = 53


⇒ x = 5



Question 39.

Solve the equation in each of the following.



Answer:


⇒ 


⇒ 


(∵ loga(M ÷ N) = logaM - logaN)


⇒ 


(∵ ax = b is the exponential form of logarithmic form logab)


⇒ 


⇒ 


⇒ 5x - 2 = 3(x + 4)


⇒ 5x - 2 = 3x + 12


⇒ 5x - 3x = 12 + 2


⇒ 2x = 14


⇒ x = 7



Question 40.

Given loga2 = x, loga 3 = y and loga 5 = z. Find the value in each of the following in terms of x, y and z.

(i) loga15 (ii) loga8 (iii) loga30

(iv)  (v)  (vi) loga1.5


Answer:

(i) loga15 = loga(5×3)


i.e. loga(5×3) = loga5 + loga3


(∵ loga(M×N) = (logaM) + (logaN))


= z + y(∵ loga5 = z,loga3 = y)


(ii) loga8 = loga23 = 3loga2 = 3x


(∵ loga2 = x)


(iii) loga30 = loga(5×3×2) = loga(5) + loga(3) + loga(2)


(∵ loga(M×N) = (logaM) + (logaN))


= z + y + x


(∵ loga5 = z,loga3 = y,loga2 = x)


= x + y + z


(iv) 


⇒ loga(3×3×3) - loga(5×5×5)


⇒ (loga3 + loga3 + loga3) - (loga5 + loga5 + loga5)


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


(v) 


⇒ loga10 - loga3


(∵ loga(M ÷ N) = logaM - logaN)


Here loga10 = loga(5×2)


(∵ loga(M×N) = (logaM) + (logaN))


= loga5 + loga2 = z + x (∵ loga5 = z,loga2 = x)


(vi) 


⇒ 


(∵ loga(M ÷ N) = (logaM) - (logaN))


= y - x(∵ loga3 = y,loga2 = x)



Question 41.

Prove the following equations.

log101600 = 2 + 4log102


Answer:

log101600 = 2 + 4log102 = 2log1010 + 4log102


Let us consider the RHS:


i.e. 2 + 4log102 = 2log1010 + 4log102


(∵ logaa = 1)


= log10102 + log1024


(∵ logaMn = nlogaM)


= log10100 + log1016


= log10(100×16)


(∵ loga(M×N) = (logaM) + (logaN))


= log101600


Hence LHS = RHS



Question 42.

Prove the following equations.

log1012500 = 2 + 3log105


Answer:

log1012500 = 2 + 3log105 = 2log1010 + 3log105


Let us consider the RHS:


i.e. 2 + 3log105 = 2log1010 + 3log105


= log10102 + log1053


(∵ logaMn = nlogaM)


= log10(102×53)


(∵ loga(M×N) = (logaM) + (logaN))


= log10(100×125)


= log10(12500)


Hence LHS = RHS



Question 43.

Prove the following equations.

log102500 = 4 - 2log102


Answer:

log102500 = 4 - 2log102


Let us consider the RHS:


i.e. 4 - 2log102 = 4log1010 - 2log102


= log10104 - log1022


(∵ logaMn = nlogaM)



(∵ loga(M ÷ N) = (logaM) - (logaN))


Hence LHS = RHS



Question 44.

Prove the following equations.

log100.16 = 2log104 – 2


Answer:

log100.16 = 2log104 - 2


Let us consider the RHS:


i.e. 2log104 - 2 = 2log104 - 2log1010


= log1042 - log10102


(∵ logaMn = nlogaM)



(∵ loga(M ÷ N) = (logaM) - (logaN))


= log10(0.16) = log100.16


Hence LHS = RHS



Question 45.

Prove the following equations.

log50.00125 = 3 - 5log510


Answer:

log50.00125 = 3 - 5log510


Let us consider the RHS:


i.e. 3 - 5log510 = 3log55 - 5log510(∵ logaa = 1)


= log553 - log5105


(∵ logaMn = nlogaM)



(∵ loga(M ÷ N) = (logaM) - (logaN))


= log50.00125



Question 46.

Prove the following equations.



Answer:


Let us consider the RHS





(∵)


= log56 - log52 + 4


= log56 - log52 + 4log55



(∵ loga(M ÷ N) = ( logaM) - (logaN) and loga(M×N) = (logaM ) + (logaN))


= log51875


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 × 101



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 × 103



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 105, we get





Thus, scientific notation of 924300 = 9.243 × 105



Question 5.

Write each of the following in scientific notation:

0.009243


Answer:

Let N = 0.009243


Divide N by 106 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 105 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 × 103


Consider,


log 4576 = log (4.576 × 103 )


= log 4.576 + log 103


(since, log (a×b) = log a + log b)


= log 4.576 + 3 (since, log 10n = 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 × 101


Consider,


log 24.56 = log (2.456 × 101 )


= log 2.456 + log 101


(since, log (a×b) = log a + log b)


= log 2.456 + 1 (since, log 10n = 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 105 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 10n = 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 104 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 10n = 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 104 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 10n = 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 × 100 )


= log 6.453 + log 100


(since, log (a×b) = log a + log b)


= log 6.453 + 0


(since, log 10n = 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 × 104


Consider,


log 23750 = log (2.3750 × 104 )


= log 2.375 + log 104


(since, log (a×b) = log a + log b)


= log 2.375 + 4


(since, log 10n = 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 × 101


Consider,


log 23.75 = log (2.375 × 101 )


= log 2.375 + log 101


(since, log (a×b) = log a + log b)


= log 2.375 + 1


(since, log 10n = 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 × 100 )


= log 2.375 + log 100


(since, log (a×b) = log a + log b)


= log 2.375 + 0


(since, log 10n = 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 10n = 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 107, we get





Thus, scientific notation 23750000 = 2.375 × 107


Consider,


log 23750000 = log (2.375 × 107 )


= log 2.375 + log 107


(since, log (a×b) = log a + log b)


= log 2.375 + 7


(since, log 10n = 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 108 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 10n = 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 × 101


Consider,


log 23.17 = log (2.317 × 101 )


= log 2.317 + log 101


(since, log (a×b) = log a + log b)


= log 2.317 + 1


(since, log 10n = 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 × 100 )


= log 9.321 + log 100


(since, log (a×b) = log a + log b)


= log 9.321 + 0


(since, log 10n = 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 × 102


Consider,


log 329.5 = log (3.295 × 102 )


= log 3.295 + log 102


(since, log (a×b) = log a + log b)


= log 3.295 + 2


(since, log 10n = 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 106 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 10n = 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 104 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 10n = 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 × 103


Consider,


log 6576 = log (6.576 × 103 )


= log 6.576 + log 103


(since, log (a×b) = log a + log b)


= log 6.576 + 3


(since, log 10n = 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 an = 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 an = 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 an = 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 an = 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 an = 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:

log9 63.28


Answer:

Let log9 63.28 = log1063.28 × log910


(since, logaM = logbM × logab)





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:

log3 7


Answer:

Let log3 7 = log107× log310


(since, logaM = logbM × logab)





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 4510 to base 2


Answer:


Thus, 4510 = 1011012



Question 2.

Convert 7310 to base 2.


Answer:


Thus, 7310 = 10010012



Question 3.

Convert 11010112 to base 10.


Answer:

11010112 = 1 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 0 × 22 +


1 × 21+ 1 × 20


= 64 + 32+ 0+8+0+2+1 = 10710


Thus, 11010112= 10710



Question 4.

Convert 1112 to base 10.


Answer:

1112 = 1 × 22 + 1 × 21 +1 × 20


= 4 + 2+1 = 710


Thus, 1112=710



Question 5.

Convert 98710 to base 5.


Answer:


Thus, 98710 = 124225



Question 6.

Convert 123810 to base 5.


Answer:


Thus, 123810 = 144235



Question 7.

Convert 102345 to base 10.


Answer:

102345 = 1 × 54 + 0 × 53 + 2 × 52 + 3 × 51+ 4 × 50


= 625 + 0 + 50+15+4 = 69410


Thus, 102345 = 69410



Question 8.

Convert 2114235 to base 10.


Answer:

2114235 = 2 × 55 + 1 × 54 + 1 × 53 + 4 × 52 + 2 × 51+


3 × 50


= 6250 + 625+ 125+100+10+3 = 711310


Thus, 2114235 = 711310



Question 9.

Convert 9856710 to base 8.


Answer:


Thus, 9856710 = 3004078



Question 10.

Convert 68810 to base 8.


Answer:


Thus, 68810 = 12608



Question 11.

Convert 471568 to base 10.


Answer:

471568 = 4 × 84 + 7 × 83 + 1 × 82 + 5 × 81 + 6 × 80


= 16384+3584+64+40+6 = 2007810


Thus, 471568 = 2007810



Question 12.

Convert 58510 to base 2,5 and 8.


Answer:


Thus, 58510 = 10010010012



Thus, 58510 = 43205



Thus, 58510 = 11118




Exercise 2.5
Question 1.

The scientific notation of 923.4 is
A. 9.234 × 10–2

B. 9.234 × 102

C. 9.234 × 103

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 × 102


Correct answer is (B)


Question 2.

The scientific notation of 0.00036 is
A. 3.6 × 10–3

B. 3.6 × 103

C. 3.6 × 10–4

D. 3.6 × 104


Answer:

Let N = 0.00036


Divide N by 105 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 103is
A. 257

B. 2570

C. 25700

D. 257000


Answer:

2.57 x 103 = 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 52 = 25 is
A. log52 = 25

B. log25 = 25

C. log525 =2

D. log255 = 2


Answer:

We know that x = logab is the logarithmic form of the exponential form b = ax


Thus, here exponential form 52 = 25 is given


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


Thus, its logarithmic form is 2 = log5 25


Hence, correct answer is (C)


Question 6.

The exponential form of log216 = 4 is
A. 24 = 16

B. 42 = 16

C. 216 = 4

D. 416 = 2


Answer:

We know that x = logab is the logarithmic form of the exponential form b = ax


Thus, here logarithmic form log216 = 4 is given


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


Thus, its logarithmic form is 24 = 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 = log497


Thus, its exponential form is


⇒ 49x = 7


⇒ (72)x = 7


⇒ 72x = 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.

log108 + log105– log104 =
A. log109

B. log1036

C. 1

D. – 1


Answer:

Consider, log108 + log105– log104 = log10(8× 5) – log104


(since, logaM + logaN = loga(M× N) )


⇒ log108 + log105– log104 = log10(40) – log104


= log10(40 ÷ 4)


(since, logaM – logaN = loga(M÷N) )


⇒ log108 + log105– log104 = log10(10) = 1 (since, logaa = 1)


Thus, correct answer is (C)


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