Converting Between Exponential and Logarithmic Forms

If a and b are positive real numbers with b≠1, the logarithm of a with base b is the number c such that b raised to the power c is equal to a. $$ \log_b(a)=c \Leftrightarrow b^c=a $$

For the natural logarithm, the relationship becomes

$$ \ln(a)=\log_e(a)=c \Leftrightarrow e^c=a $$

A Practical Example

Example 1 ( converting from exponential form to logarithmic form )

Consider the exponential equation

$$ 3^x = 8 $$

Since 3 cannot be written as a power of 8, the equation is rewritten in logarithmic form to solve for x.

$$ x = \log_3(8) $$

Example 2 ( converting from logarithmic form to exponential form )

Consider the logarithmic expression

$$ x = \log_3(9) $$

To evaluate the logarithm, rewrite it as an exponential equation.

$$ 3^x = 9 $$

Written in this form, it becomes immediately clear that the solution is x=2.

$$ 3^2 = 9 $$

And the same procedure applies in general.