HOW TO EVALATE LOGARITHMS WITH FRACTIONS

To evaluate logarithmic function, we should know how to convert the logarithmic function into exponential form.

Evaluate each of the following logarithms without the use of a calculator.

Problem 1 :

log₄ 1/2 =

Solution :

log₄ 1/2 = x

4^x = 1/2

(2²)^x = 1/2

(2)^2x = 2^-1

2x = -1

x = -1/2

Problem 2 :

log₆ 1/36 =

Solution :

log₆ 1/36 = x

6^x = 1/36

6^x = 1/6²

6^x = 6^-2

x = -2

Problem 3 :

log_2/3 9/4 =

Solution :

log_2/3 9/4 = x

(2/3)^x = 9/4

(2/3)^x = (3/2)²

(2/3)^x = (2/3)^-²

x = -2

Problem 4 :

log₈₁ 1/27 =

Solution :

log₈₁ 1/27 = x

81^x = 1/27

(3⁴)^x = 1/3³

(3)^4x = 3^-3

4x = -3

x = -3/4

Problem 5 :

log₄ 1/8 =

Solution :

log₄ 1/8 = x

4^x = 1/8

(2²)^x = 1/2³

(2)^2x = 2^-3

2x = -3

x = -3/2

Problem 6 :

log₃₆ 1/6 =

Solution :

log₃₆ 1/6 = x

36^x = 1/6

(6²)^x = 1/6

(6)^2x = 6^-1

2x = -1

x = -1/2

Problem 7 :

log_1/16 32 =

Solution :

log_1/16 32 = x

(1/16)^x = 32

(1/2⁴)^x = 2⁵

(2)^-4x = 2⁵

-4x = 5

x = -5/4

Problem 8 :

log₂₇ 1/3 =

Solution :

log₂₇ 1/3 = x

27^x = 1/3

(3³)^x = 1/3

(3)^3x = 3^-1

3x = -1

x = -1/3

Problem 9 :

log_1/4 16 =

Solution :

log_1/4 16 = x

(1/4)^x = 16

(1/2²)^x = 2⁴

(2)^-2x = 2⁴

-2x = 4

x = -4/2

x = -2