How to Use the Power Rule for Logarithms

Problem

\log_2(8^3)

Answer

9

The power rule lets you pull an exponent out of a logarithm. Multiply the exponent by the log of the base number.

Step-by-step solution

Write the power rule.
$\log_b(M^p) = p\log_b M$
Why: The exponent moves to the front as a multiplier.
Move the 3 in front.
$3\log_2 8 = 3 \times 3$
Why: log base 2 of 8 is 3 because 2 cubed is 8.
Multiply.
$3 \times 3 = 9$
Why: The original number is 512, which is 2 to the ninth power.

Adding the exponent to the log instead of multiplying: 3 + 3 = 6.
Applying the rule to the base, not the argument.
Computing 8 cubed first and making an arithmetic slip. Move the exponent out first.

Use the same method. Work each problem on paper, then open the answer to check.

$\log_2(4^3)$

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$6$
$\log_3(3^4)$

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$4$
$\log_{10}(10^5)$

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$5$
$\log_2(2^7)$

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$7$
$\log_5(5^4)$

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$4$