Convert between exponential and logarithmic form

The relationship $a^b=c \Longleftrightarrow \log_a(c)=b$ is understood for $a>0$ , $a\ne1$ , and $c>0$ : the base remains the logarithm’s base, the exponent becomes the logarithm’s value, and the exponential result becomes its argument. This inverse relationship supports interpreting and solving exponential and logarithmic equations; the scope is limited to real-valued expressions and does not include complex logarithms or more advanced logarithmic generalizations.

Detailed Explanation: 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.

The base is $2$ , so the logarithm’s base is also $2$ .
The result is $32$ , so $32$ goes inside the logarithm.
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 $c>0$ .

Learn by doing: Convert between exponential and logarithmic form

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Logarithms - Convert Exponent to Logarithm - Fraction Value

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