Convert Logarithms and Exponentials

The logarithmic functions logb x and the exponential functions bx are inverse of each other, hence

y = logb x is equivalent to x = by


where b is the common base of the exponential and the logarithm.

The above equivalence helps in solving logarithmic and exponential functions and needs a deep understanding. Examples, of how the above relationship between the logarithm and exponential may be used to transform expressions, are presented below.

Example 1 : Change each logarithmic expression to an exponential expression.

1. log3 27 = 3

2. log36 6 = 1 / 2

3. log2 (1 / 8) = -3

4. log8 2 = 1 / 3

Solution to Example 1:

1.     The logarithmic form log3 27 = 3 is equivalent to the exponential form

27 = 33


2.     The logarithmic form log36 6 = 1 / 2 is equivalent to the exponential form
6 = 361/2


3.     log2 (1 / 8) = -3 in exponential form is given by

1 / 8 = 2-3


4.     log8 2 = 1 / 3 in exponential form is given by

2 = 81/3



Example 2 : Change each exponential expression to logarithmic expression.

1. 34 = 81

2. 41/2 = 2

3. 3-3 = 1 / 27

4. 103 = 1000

Solution to Example 2:

1.     The exponential form 34 = 81 is equivalent to the logarithmic form

4 = log3 81


2.     The exponential form 41/2 = 2 is equivalent to the logarithmic form
1 / 2 = log4 2


3.     3-3 = 1 / 27 in logarithmic form is given by

-3 = log3 (1/ 27)


4.     103 = 1000 in logarithmic form is given by

3 = log10 1000

Example 3 : Solve for x the following equations.

1. log3 x = 5

2. log2 (x - 3) = 2

3. 2 log3 (- x + 1) = 6

Solution to Example 3:

1.     To solve the equation log3 x = 5, rewrite it into exponential form

x = 35


2.     Rewrite the equation log2 (x - 3) = 2 into exponential form
x - 3 = 22 = 4
Solve for x

x = 4 + 3 = 7

3.     Divide all terms of the equation 2 log3 (- x + 1) = 6 by 2
log3 (- x + 1) = 3


Rewrite the equation obtained in exponential form

- x + 1 = 33 = 27

Solve for x

x = - 26




More references and links related to the logarithmic functions.
Solve Exponential and Logarithmic Equations (self test).


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Updated: 2 April 2013

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