Skill: Apply the power law of logarithms

Explanation and Free Practice Resources

For a logarithm with base $b>0$ , $b\ne1$ , and positive $x$ , the power law $\log_b(x^p)=p\log_b(x)$ expresses how logarithms convert exponents into multiplicative factors, supporting symbolic simplification and solving exponential equations. At this level, it applies to familiar integer and rational powers with valid positive domains; it does not include complex-valued logarithms or more advanced generalizations.

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Detailed Explanation: Apply the power law of logarithms

The power law of logarithms lets you move an exponent from inside a logarithm to the front:

\log_b(x^p)=p\log_b(x)

Here, $b>0$ , $b\ne 1$ , and $x>0$ . The exponent becomes a multiplier; it does not stay inside the logarithm.

Worked example

Simplify

\log_7\left((x^2)^3\right), \qquad x>0.

Step 1: Apply the power law to the outer exponent.

\log_7\left((x^2)^3\right)=3\log_7(x^2)

Step 2: Apply the power law again to $x^2$ .

3\log_7(x^2)=3\left(2\log_7 x\right)

Step 3: Multiply the numerical factors.

3\left(2\log_7 x\right)=6\log_7 x

Therefore,

\boxed{\log_7\left((x^2)^3\right)=6\log_7 x}

The two exponents, $3$ and $2$ , become factors in front of the logarithm.

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Logarithms - Power Property - Power To Product (Integers)

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