Skill: Convert between exponential and logarithmic form

An exponential equation and a logarithmic equation can say the same thing in two different ways:

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

The base stays the same, the result of the exponentiation becomes the logarithm’s argument, and the exponent becomes the logarithm’s value.

For these equations, the conditions are

$$a>0,\qquad a\ne 1,\qquad c>0.$$

Worked example

Rewrite the exponential equation

$$2^5=32$$

in logarithmic form.

Step 1: Identify the parts.

In $2^5=32$:

• The base is $2$.
• The exponent is $5$.
• The result is $32$.

Step 2: Use the conversion pattern.

Since

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

substitute $a=2$, $b=5$, and $c=32$:

$$\boxed{\log_2(32)=5}$$

This means “the exponent to which $2$ must be raised to get $32$ is $5$.”

The conversion also works in reverse. Starting with

$$\log_2(32)=5,$$

rewrite it as

$$\boxed{2^5=32}.$$

Always check that the base is positive and not $1$, and that the logarithm’s argument is positive.