CONVERTING BETWEEN LOGARITHMIC AND EXPONENTIAL FORMS WORKSHEET

Problem 1 : 

Convert the following into exponential form : 

log₄64  =  3

Problem 2 : 

Convert the following into exponential form : 

log₅(1/25)  =  -2

Problem 3 : 

Convert the following into exponential form : 

log_√39  =  4

Problem 4 : 

Convert the following into exponential form : 

log₁₀0.1 =  -1

Problem 5 : 

Convert the following into exponential form : 

log_0.58  =  -3

Problem 6 : 

Convert the following into logarithmic form : 

1/1296  =  6^-4

Problem 7 : 

Convert the following into logarithmic form : 

 (13)^-1  =  1/13

Problem 8 : 

Convert the following into logarithmic form : 

 8^-2/3  =  1/4

Problem 9 : 

Convert the following into logarithmic form : 

10^-1  =  0.1

Problem 10 :

Solve for x :

log_1/5x  =  3

Problem 11 :

Solve for x :

log_x125√5  =  7

Problem 12 :

Solve for x :

log_x0.001  =  -3

Problem 13 :

Solve for x : 

log₅(5log₃x)  =  2

Problem 14 :

Solve for x : 

x + 2log₂₇9  =  0

Problem 15 :

Solve for x :

log₃x + log₉x + log₈₁x  =  7/4

Detailed Answer Key

Problem 1 : 

Convert the following into exponential form : 

log₄64  =  3

Solution : 

64  =  4³

Problem 2 : 

Convert the following into exponential form : 

log₅(1/25)  =  -2

Solution : 

1/25  =  5^-2

Problem 3 : 

Convert the following into exponential form : 

log_√39  =  4

Solution : 

9  =  (√3)⁴

Problem 4 : 

Convert the following into exponential form : 

log₁₀0.1 =  -1

Solution : 

0.1  =  10^-1

Problem 5 : 

Convert the following into exponential form : 

log_0.58  =  -3

Solution : 

8  =  0.5^-3

Problem 6 : 

Convert the following into logarithmic form : 

1/1296  =  6^-4

Solution : 

log₆(1/1296)  =  -4

Problem 7 : 

Convert the following into logarithmic form : 

 (13)^-1  =  1/13

Solution :

-1  =  log₁₃(1/13)

Problem 8 : 

Convert the following into logarithmic form : 

 8^-2/3  =  1/4

Solution :

-2/3  =  log₈(1/4)

Problem 9 : 

Convert the following into logarithmic form : 

10^-1  =  0.1

Solution :

-1  =  log₁₀(0.1)

Problem 10 :

Solve for x :

log_1/5x  =  3

Solution :

log_1/5x  =  3

Convert to exponential form. 

x  =  (1/5)³

x  =  1³/5³

x  =  1/125

Problem 11 :

Solve for x :

log_x125√5  =  7

Solution :

log_x125√5  =  7

Convert to exponential form. 

125√5  =  x⁷

 5 ⋅ 5 ⋅ 5 ⋅ √5  =  x⁷

Each 5 can be expressed as (√5 ⋅ √5).

Then,

√5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5  =  x⁷

√5⁷  =  x⁷

Because the exponents are equal, bases can be equated. 

x  =  √5

Problem 12 :

Solve for x :

log_x0.001  =  -3

Solution :

log_x0.001  =  -3

Convert to exponential form. 

0.001  =  x^-3

1/1000  =  1/x³

Take reciprocal on both sides. 

1000  =  x³

10³  =  x³

Because the exponents are equal, bases can be equated. 

10  =  x

Problem 13 :

Solve for x : 

log₅(5log₃x)  =  2

Solution :

log₅(5log₃x)  =  2

Convert to exponential form. 

5log₃x  =  5²

5log₃x  =  25

Divide each side by 5.

log₃x  =  5

Convert to exponential form. 

x  =  3⁵

x  =  243

Problem 14 :

Solve for x : 

x + 2log₂₇9  =  0

Solution :

x + 2log₂₇9  =  0

x  =  -2log₂₇9

x  =  log₂₇9^-2

Convert to exponential form. 

27^x  =  9^-2

(3³)^x  =  (3²)^-2

3^3x  =   3^-4

Because the bases are equal, exponents can be equated. 

3x  =  -4

x  =  -4/3

Problem 15 :

Solve for x :

log₃x + log₉x + log₈₁x  =  7/4

Solution :

log₃x + log₉x + log₈₁x  =  7/4

(1 / log_x3)  +  (1 / log_x9)  +  (1 / log_x81)  =  7/4

(1 / log_x3)  +  (1 / log_x9)  +  (1 / log_x81)  =  7/4

(1 / log_x3)  +  (1 / log_x3²)  +  (1 / log_x3⁴)  =  7/4

(1 / log_x3)  +  (1 / 2log_x3)  +  (1 / 4log_x3)  =  7/4

(4 / 4log_x3)  +  (2 / 4log_x3)  +  (1 / 4log_x3)  =  7/4

(4 + 2 + 1) / 4log_x3  =  7/4

7 / 4log_x3  =  7/4

Multiply each side by 4/7.

1 / log_x3  =  1

log₃x  =  1

Convert to exponential form. 

x  =  3¹

x  =  3

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