# Convert Logarithms and Exponentials

## Relationship Between Exponential and Logarithm

The logarithmic functionslogb x and the exponential functionsbx 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. log
3 27 = 3
2. log
36 6 = 1 / 2
3. log
2 (1 / 8) = -3
4. log
8 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 log
36 6 = 1 / 2 is equivalent to the exponential form
6 = 361/2

3.     log
2 (1 / 8) = -3 in exponential form is given by
1 / 8 = 2-3

4.     log
8 2 = 1 / 3 in exponential form is given by
2 = 81/3

### Example 2

Change each exponential expression to logarithmic expression.
1. 3
4 = 81
2. 4
1/2 = 2
3. 3
-3 = 1 / 27
4. 10
3 = 1000

Solution to Example 2:
1.     The exponential form 34 = 81 is equivalent to the logarithmic form

4 = log3 81

2.     The exponential form 4
1/2 = 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.     10
3 = 1000 in logarithmic form is given by
3 = log10 1000

### Example 3

Solve for x the following equations.
1. log
3 x = 5
2. log
2 (x - 3) = 2
3. 2 log
3 (- 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 = 3
3 = 27
Solve for x
x = - 26

## More References and Links Related to the Logarithmic Functions

Solve Exponential and Logarithmic Equations (self test).
Rules of Logarithms and Exponentials - Questions with Solutions.