Convert between exponential and logarithmic form

The equivalence $a^b=c \iff \log_a(c)=b$ expresses logarithms as the inverse operation of exponentiation: the logarithm gives the exponent to which a positive base must be raised to produce a given positive value. This includes translating among exponential equations, logarithmic equations, and function representations while respecting $a>0$ , $a\ne1$ , and $c>0$ , and supports solving exponential and logarithmic equations; complex logarithms and more abstract extensions are not included.

Detailed Explanation: 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.

Learn by doing: Convert between exponential and logarithmic form

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Logarithms - Convert Exponent to Logarithm - Fraction Value

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