Skip to main content
Mathovia

Math Practice

Logarithm Rules Practice Problems

The three log rules turn products into sums, quotients into differences and powers into multiples. These twelve problems use them in both directions — combining logs to evaluate them exactly, expanding a single log into pieces, and condensing several logs back into one.

  1. 1.

    Evaluate .

    Show full solution

    Either evaluate each () or use the product rule: .

  2. 2.

    Evaluate .

    Show full solution

    Quotient rule: .

  3. 3.

    Evaluate .

    Show full solution

    Power rule: .

  4. 4.

    Evaluate .

    Show full solution

    Product rule: . Neither log is a nice number alone, but their sum is.

  5. 5.

    Evaluate .

    Show full solution

    Product rule: , since .

  6. 6.

    Evaluate .

    Show full solution

    Quotient rule: .

  7. 7.

    Evaluate .

    Show full solution

    Write the root as a power: .

  8. 8.

    Evaluate .

    Show full solution

    Product rule: .

  9. 9.

    Given and , find to four decimal places.

    Show full solution

    , so .

  10. 10.

    Expanding gives . Enter , and , separated by commas.

    Show full solution

    Quotient rule, then product rule, then power rule: . So , , .

  11. 11.

    Condense into . Enter .

    Show full solution

    Coefficients become exponents, sums become products and differences become quotients: .

  12. 12.

    Condense into . Enter .

    Show full solution

    Power rule first: . Then combine: .
Written by
MPhil in Mathematics, Founder of Mathovia
Published

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Logarithm Rules Practice Problems

The rules below are derived in the logarithm rules lesson. Keep the table in view while you work — every problem uses at least one of them.

The rules

RuleStatement
Product
Quotient
Power

Expanding and condensing

Expanding reads the rules left to right; condensing reads them right to left. When condensing, move every coefficient into an exponent before you combine anything.

Traps to avoid

cannot be split, is not , and logs with different bases cannot be combined without first changing the base.

Common mistakes to watch for

  • Splitting a log of a sum. cannot be split; only products and quotients can.
  • Misusing the power rule. , but stays as it is.
  • Wrong quotient order. .
  • Mixing bases. The rules only combine logs with the same base.

Frequently Asked Questions

What are the three main log rules?+

Product: log(MN) = log M + log N. Quotient: log(M/N) = log M − log N. Power: log(Mᵖ) = p·log M. They hold for every base, including log and ln.

Can I split log(x + y)?+

No. There is no rule for the log of a sum. The product rule is for log(xy), and log(x + y) has to stay as it is.

Is log M ÷ log N the same as log(M/N)?+

No. The quotient rule needs the division inside a single log. A quotient of two logs is a change of base: log 8 ÷ log 2 = log₂ 8 = 3.

What order should I use when condensing?+

Apply the power rule first, so every coefficient becomes an exponent. Then turn sums into products and differences into quotients until one log remains.

How do I type a condensed answer?+

Type the expression inside the log using ^ for powers and / for division, for example x^2*y^3/z. Any equivalent form is accepted.

More practice problems

Logarithm Practice Problems Change of Base Practice Problems Solving Logarithmic Equations Practice Problems