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 log_{3} 27 = 3 is equivalent to the exponential form

27 = 3^{3}

2. The logarithmic form log_{36} 6 = 1 / 2 is equivalent to the exponential form

6 = 36^{1/2}

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 = 8^{1/3}

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 3^{4} = 81 is equivalent to the logarithmic form

4 = log_{3} 81

2. The exponential form 4^{1/2} = 2 is equivalent to the logarithmic form

1 / 2 = log_{4} 2

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

-3 = log_{3} (1/ 27)

4. 10^{3} = 1000 in logarithmic form is given by

3 = log_{10} 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 log_{3} x = 5, rewrite it into exponential form

x = 3^{5}

2. Rewrite the equation log_{2} (x - 3) = 2 into exponential form

x - 3 = 2^{2} = 4

Solve for x

x = 4 + 3 = 7

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