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

Class 9^th 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

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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

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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) log_aM - N = log_aMlog_aN

Answer:

(i) True

log₅125 = 3

⇒ 5³ = 125

(∵ x = log_ab is the logarithmic form of the exponential form a^x = b)

This is true.

(ii) False

⇒ 

(∵ x = log_ab 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.

(∵ log_aM ÷ log_aN = log_aM - log_aN

;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_ab 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_aN

Let us consider the RHS

log_aM ÷ log_aN = log_aM - log_aN

(∵ according to quotient rule,log_aM ÷ log_aN = log_aM - log_aN ;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.

2⁴ = 16

Answer:

Here it’s given that 2⁴ = 16,

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

log_ab is the logarithmic form of the exponential form a^x = 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 a^x = b.

log_ab is the logarithmic form of the exponential form a^x = 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 a^x = b.

log_ab 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_ab 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_ab 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_ab 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.

log₆216 = 3

Answer:

Here it’s given that log₆216 = 3

The given equation is in the form of log_ab = x

The exponential form of the logarithmic form log_ab is a^x = 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 log_ab = x

The exponential form of the logarithmic form log_ab is a^x = 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 log_ab = x

The exponential form of the logarithmic form log_ab is a^x = 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 log_ab = x

The exponential form of the logarithmic form log_ab 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_ab = x

The exponential form of the logarithmic form log_ab is a^x = b.

In the given equation  ( a = 64,,)

⇒ 

Question 13.

Obtain the equivalent exponential form of the following.

log_0.58 = - 3

Answer:

Here it’s given that log_0.58 = - 3

The given equation is in the form of log_ab = x

The exponential form of the logarithmic form log_ab is a^x = b.

In the given equation log_0.58 = - 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)

(∵ nlog_aM = log_aMⁿ)

⇒ - 4(1) = - 4

(log_aa = 1)

Question 15.

Find the value of the following

log₇ 343

Answer:

log₇343 = log₇7³

⇒ 3log₇7 (∵ nlog_aM = log_aMⁿ)

⇒ 1(∵ log_aa = 1)

Question 16.

Find the value of the following

log₆6⁵

Answer:

log₆6⁵

⇒ 5log₆6

(∵ nlog_aM = log_aMⁿ)

= 5(1)

(∵ log_aa = 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_ab)

⇒ 3( - 1) = - 3

Question 18.

Find the value of the following

log₁₀ 0.0001

Answer:

Here we have log100.0001, i.e.

⇒ 

⇒ - 4log₁₀10 (∵ nlog_aM = log_aMⁿ)

⇒ - 4(1) = - 4 (∵ log_aa = 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_ab)

⇒ 

⇒ 

⇒ 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_ab)

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.

log_x 0.001 = – 3

Answer:

log_x0.001 = - 3

⇒ x ^{- 3} = 0.001

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

⇒ 

⇒ 

⇒ x = 10

Question 25.

Solve the following equations.

x + 2 log₂₇ 9 = 0

Answer:

x + 2log₂₇9 = 0

⇒ x = - 2log₂₇9

⇒ x = log₂₇9 ^{- 2}

⇒ x = log₃³(3²) ^{- 2}

⇒ x = log₃³(3) ^{- 4}

⇒ (3³)^x = 3 ^{– 4}

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

⇒ 3x = - 4 (compare the exponents)

⇒ 

Question 26.

Simplify the following.

log₁₀3 + log₁₀3

Answer:

log₁₀3 + log₁₀3 = log₁₀(3×3) = log₁₀9

(∵ using the product rule,log_a(M×N) = (log_aM) + (log_aN);a,M,N are positive numbers ,a≠1)

Question 27.

Simplify the following.

log₂₅35 – log₂₅10

Answer:

(using the quotient rule log_a(M ÷ N) = (log_aM) - (log_aN) );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.

log_a(M×N) = (log_aM) + (log_aN) and

log_a(M ÷ N) = (log_aM) - (log_aN))

⇒ 

⇒ 

⇒ log₇7²

⇒ 2log₇7 = 2

(∵ log₇7 = 1 )

Question 29.

Simplify the following.

Answer:

⇒ log₈(16×52) - log₈13

(∵ log_a(M×N) = (log_aM) + (log_aN) and )

⇒ 

(∵ loga(M ÷ N) = (log_aM) - (log_aN))

⇒ log₈(16×4) = log₈64

⇒ 8x = 64 or x = 2

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

Question 30.

Simplify the following.

5log₁₀2 + 2log₁₀3 - 6log₆₄4

Answer:

5log₁₀2 + 2log₁₀3 - 6log₆₄4

Here log₆₄4 = x

⇒ 64^x = 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

(∵ log_a(M×N) = (log_aM) + (log_aN))

⇒ 

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

⇒ log10(2×5) = log1010 = 1

(∵ log_aa = 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

⇒ 

(∵)

⇒ 

(∵ log_aMⁿ = nlog_aM)

⇒ 

⇒ 

⇒ 

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

(∵ a^x = b is the exponential form of logarithmic form of log_ab)

⇒ log₂x⁸ = 16

(∵ log_aMⁿ = nlog_aM)

⇒ 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

⇒ 

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

⇒ 

⇒ 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:

⇒ 

⇒ 

(∵ log_a(M ÷ N) = log_aM - log_aN)

⇒ 

(∵ a^x = b is the exponential form of logarithmic form log_ab)

⇒ 

⇒ 

⇒ 5x - 2 = 3(x + 4)

⇒ 5x - 2 = 3x + 12

⇒ 5x - 3x = 12 + 2

⇒ 2x = 14

⇒ x = 7

Question 40.

Given log_a2 = 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_a15 (ii) log_a8 (iii) log_a30

(iv)  (v)  (vi) log_a1.5

Answer:

(i) log_a15 = log_a(5×3)

i.e. log_a(5×3) = log_a5 + log_a3

(∵ log_a(M×N) = (log_aM) + (log_aN))

= z + y(∵ log_a5 = z,log_a3 = y)

(ii) log_a8 = log_a2³ = 3log_a2 = 3x

(∵ log_a2 = x)

(iii) log_a30 = log_a(5×3×2) = log_a(5) + log_a(3) + log_a(2)

(∵ log_a(M×N) = (log_aM) + (log_aN))

= z + y + x

(∵ log_a5 = z,log_a3 = y,log_a2 = x)

= x + y + z

(iv) 

⇒ log_a(3×3×3) - log_a(5×5×5)

⇒ (log_a3 + log_a3 + log_a3) - (log_a5 + log_a5 + log_a5)

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

(v) 

⇒ log_a10 - log_a3

(∵ log_a(M ÷ N) = log_aM - log_aN)

Here log_a10 = log_a(5×2)

(∵ log_a(M×N) = (log_aM) + (log_aN))

= log_a5 + log_a2 = z + x (∵ log_a5 = z,log_a2 = x)

(vi) 

⇒ 

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

= y - x(∵ log_a3 = y,log_a2 = 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

(∵ log_aa = 1)

= log₁₀10² + log₁₀2⁴

(∵ log_aMⁿ = nlog_aM)

= log₁₀100 + log₁₀16

= log₁₀(100×16)

(∵ log_a(M×N) = (log_aM) + (log_aN))

= 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

(∵ log_aMⁿ = nlog_aM)

= log₁₀(10²×5³)

(∵ log_a(M×N) = (log_aM) + (log_aN))

= 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

(∵ log_aMⁿ = nlog_aM)

= 

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

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

(∵ log_aMⁿ = nlog_aM)

= 

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

= 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(∵ log_aa = 1)

= log₅5³ - log₅10⁵

(∵ log_aMⁿ = nlog_aM)

= 

(∵ log_a(M ÷ N) = (log_aM) - (log_aN))

= log₅0.00125

Question 46.

Prove the following equations.

Answer:

Let us consider the RHS

(∵)

₌ log₅6 - log₅2 + 4

= log₅6 - log₅2 + 4log₅5

(∵ log_a(M ÷ N) = ( log_aM) - (log_aN) and log_a(M×N) = (log_aM ) + (log_aN))

= 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, log_aM = log_bM × log_ab)

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, log_aM = log_bM × log_ab)

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 = log_ab is the logarithmic form of the exponential form b = a^x

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 = log_ab is the logarithmic form of the exponential form b = a^x

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

⇒ 49^x = 7

⇒ (7²)^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.

log₁₀8 + log₁₀5– log₁₀4 =
A. log₁₀9

B. log₁₀36

C. 1

D. – 1

Answer:

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

(since, log_aM + log_aN = log_a(M× N) )

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

= log₁₀(40 ÷ 4)

(since, log_aM – log_aN = log_a(M÷N) )

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

Thus, correct answer is (C)