Practice Logarithm Properties in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

The three fundamental rules of logarithms: the product rule \log_b(xy) = \log_b x + \log_b y, the quotient rule \log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y, and the power rule \log_b(x^n) = n\log_b x.

Logarithms were invented to turn hard operations into easy ones. Multiplication becomes addition, division becomes subtraction, and exponentiation becomes multiplication. This is why slide rules worked—they added lengths (logarithms) to multiply numbers.

Example 1

easy
Expand \log_2(8x^3) using logarithm properties.

Example 2

medium
Condense 2\ln x - \frac{1}{2}\ln(x + 1) + 3\ln y into a single logarithm.

Example 3

medium
Use the change of base formula to evaluate \log_3 20 to three decimal places.

Example 4

easy
Expand \log\left(\frac{a^2}{b^5}\right).
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