Logarithmic Power Rule

July 5, 2026

A quick overview of the logarithmic power rule.

Power Rule For Logarithms

The logarithmic power rule is

\[\log_a(x^r)=r\log_a(x).\]

When you have everything inside a logarithm raised to some power $r$, you can bring the $r$ in front of the logarithm while leaving everything else inside.

Watch this section: power rule for logarithms at 0:00.

Moving A Power Outside

For example, consider

\[\log_4((x+1)^3).\]

The exponent is on the full expression $x+1$, so the $3$ moves in front of the logarithm:

\[\log_4((x+1)^3)=3\log_4(x+1).\]

It’s important that everything insider the logarithm is grouped together and raised to a power before we use the power rule.

Watch this section: moving a power outside at 0:22.

Moving A Coefficient Inside

We can also go in the other direction. Suppose we have a logarithm with a constant/coefficient in front of the logarithm like

\[15\log_{10}(y),\]

the constant/coefficient can move inside as an exponent:

\[15\log_{10}(y)=\log_{10}(y^{15}).\]

This just uses the logarithmic power rule in the opposite direction.

Watch this section: moving a coefficient inside at 0:49.

More Power-Outside Examples ($\rightarrow$ direction)

Some more quick examples where we move the power from inside the logarithm to multiplied in front of the logarithm

\[\log_5(x^{10})=10\log_5(x)\] \[\log_{12}((x+5)^y)=y\log_{12}(x+5)\] \[\ln(y^z)=z\ln(y)\]

The only thing that changed from the left to the right side of the equals sign is the position of the value that is the power.

Watch this section: multiple power-rule examples at 1:15.

More Coefficient-Inside Examples ($\leftarrow$ direction)

Some more quick examples where we move the power from in front of the logarithm to inside the logarithm

\[40\log_2(x)=\log_2(x^{40})\] \[15\log_{10}(x+y+z)=\log_{10}((x+y+z)^{15})\] \[12\ln(y)=\ln(y^{12})\]

Once again, the only thing that changed from the left to the right side of the equals sign is the position of the value that is the power.

Watch this section: moving coefficients inside at 1:25.

Timestamp Guide

Section	What is shown	Video
Power rule	Introduce $\log_a(x^r)=r\log_a(x)$.	0:00
First examples	Move the exponent in $\log_4((x+1)^3)$ outside, then move $15$ inside $15\log_{10}(y)$.	0:15
Power-out examples	Expand $\log_5(x^{10})$, $\log_{12}((x+5)^y)$, and $\ln(y^z)$.	1:15
Coefficient-in examples	Rewrite $40\log_2(x)$, $15\log_{10}(x+y+z)$, and $12\ln(y)$ with exponents inside.	1:25

Use the logarithmic power rule topic page for more power rule notes. The logarithmic power rule example archive collects the timestamped examples from this video and related JoeCMath logarithm practice.

For nearby rules, review the logarithmic product rule companion, the logarithmic quotient rule companion, or the Can You Expand These Logs? companion.

You can also watch the JoeCMath logarithms playlist: open the playlist on YouTube.