Logarithm Properties Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Logarithm Properties.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Logarithms convert between multiplicative and additive worlds. Every property follows from the fact that \log_b(b^k) = k—logarithms extract exponents.

Common stuck point: The properties only work for logs of the SAME base. You cannot combine \log_2 x + \log_3 y into a single logarithm.

Sense of Study hint: Write out what each log equals as an exponent. For example, if log_2(8) = 3, write 2^3 = 8. Then apply the property and convert back.

Worked Examples

Example 1

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

Solution

  1. 1
    Apply the product rule: \log_2(8x^3) = \log_2 8 + \log_2 x^3.
  2. 2
    Evaluate \log_2 8 = 3 and apply the power rule: \log_2 x^3 = 3\log_2 x.
  3. 3
    Result: 3 + 3\log_2 x.

Answer

3 + 3\log_2 x
The product rule \log_b(MN) = \log_b M + \log_b N and power rule \log_b(M^p) = p\log_b M are the two most commonly used expansion 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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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

These ideas may be useful before you work through the harder examples.

logarithm