Logarithmic Power Rule
This page follows the JoeCMath video Logarithmic Power Rule in 3 Mins. The video treats the logarithmic power rule as a two-way move: powers can come out of a logarithm, and coefficients can go back inside.
Power Rule For Logarithms
The video starts with the rule
\[\log_a(x^r)=r\log_a(x).\]The power $r$ begins inside the logarithm as an exponent on $x$. After using the rule, that same $r$ becomes the coefficient in front of the logarithm.
The main idea is simple: an exponent inside one logarithm can move outside as multiplication.
Watch this section: power rule for logarithms at 0:00.
Moving A Power Outside
The first worked example is
\[\log_4((x+1)^3).\]The exponent is on the full expression $x+1$, so the $3$ moves in front of the logarithm:
\[\log_4((x+1)^3)=3\log_4(x+1).\]The video keeps the parentheses around $x+1$ because the power belongs to the whole group.
Watch this section: moving a power outside at 0:22.
Moving A Coefficient Inside
The video then reverses the direction of the same rule. Starting with
\[15\log_{10}(y),\]the coefficient can move inside as an exponent:
\[15\log_{10}(y)=\log_{10}(y^{15}).\]This is not a new formula. It is the logarithmic power rule read from right to left.
Watch this section: moving a coefficient inside at 0:49.
More Power-Outside Examples
The next section shows several examples where the exponent is already inside 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 base stays the same in each example. The only thing that moves is the exponent on the logarithm’s argument.
Watch this section: multiple power-rule examples at 1:15.
More Coefficient-Inside Examples
The video also shows the reverse direction with several coefficients:
\[40\log_2(x)=\log_2(x^{40})\] \[15\log_{10}(x+y+z)=\log_{10}((x+y+z)^{15})\] \[12\ln(y)=\ln(y^{12})\]Each coefficient becomes an exponent on the entire argument of the logarithm.
Watch this section: moving coefficients inside at 1:25.
Multiple Terms Inside The Logarithm
The last point in the video focuses on a common-looking situation:
\[15\log_{10}(x+y+z).\]The expression inside the logarithm is the whole group, so the exponent belongs to the whole group:
\[15\log_{10}(x+y+z)=\log_{10}((x+y+z)^{15}).\]That is why the video’s multiple-term warning matters: the coefficient does not distribute across addition inside the logarithm.
Watch this section: multiple terms inside the logarithm at 1:36.
Timestamp Guide
Related Logarithm Work
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.