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All courses Math III · F-LE.4.2 31 of 55
Use the definition of logarithms to translate among logarithms in any base.

Rewrite a logarithmic equation in exponential form

Problem
What is the equivalent exponential form of log₂(8)=3?
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Hint

Identify the base, the number inside the logarithm, and the value after the equals sign.

The definition logb(a)=c means bc=a. The value of the logarithm becomes the exponent in the exponential form.

Solution walkthrough

01

Identify the Three Roles

\[\log_2(8)=3:~\text{base}~2,~\text{result}~8,~\text{exponent}~3\]

The subscript on log is the base, the number inside the parentheses is the result, and the value after the equals sign is the exponent.

02

Use the Log Definition

\[\log_b(a)=c~\text{means}~b^c=a\]

The definition of a logarithm says that logarithmic and exponential forms describe the same relationship.

03

Substitute the Values

\[2^3=8\]

Keep 2 as the base, use 3 as the exponent, and set the power equal to 8.

04

Check the Translation

\[2^3~=~2*2*2~=~8\]

The statement checks out because 2 multiplied by itself three times equals 8, matching the original logarithm.

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Another way

  1. Read log₂(8)=3 as the sentence, "The exponent on 2 that gives 8 is 3." That sentence translates directly to 2³=8.

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Common mistake

A common mistake is to switch the exponent and the result, writing 2⁸=3. In log₂(8)=3, the 3 is the exponent and the 8 is the result.