Skill: Convert between exponential and logarithmic form

The exponential and logarithmic forms of the same statement are connected by

$$a^b=c \quad\Longleftrightarrow\quad \log_a(c)=b.$$

To convert from exponential form to logarithmic form:

• Keep the base the same.
• Make the result the argument of the logarithm.
• Make the exponent the value of the logarithm.

Example

Convert

$$2^5=32$$

to logarithmic form.

1. The base is $2$, so the logarithm’s base is also $2$.
2. The result is $32$, so $32$ goes inside the logarithm.
3. The exponent is $5$, so $5$ goes on the other side of the equation.

Therefore,

$$2^5=32 \quad\Longleftrightarrow\quad \log_2(32)=5.$$

The reverse conversion follows the same relationship:

$$\log_2(32)=5 \quad\Longleftrightarrow\quad 2^5=32.$$

This relationship applies when $a>0$, $a\ne1$, and $c>0$.