Logarithm Rules Practice Test

About This Logarithm Rules Practice Test

This practice test isolates the algebraic properties used to expand, condense, and rewrite logarithmic expressions. It does not rely on repeating one product-rule example; the set combines several rules in the same expression.

Questions use common logarithms, natural logarithms, and general bases. Variables are assumed to lie in domains where the displayed logarithms are defined.

Skills Covered

The set covers product and quotient rules, the power rule, roots as fractional exponents, change of base, inverse identities, constants written as logarithms, and recognition of invalid rules such as log(x + y) = log x + log y.

Expansion and Condensation Strategy

1. For expansion, separate quotients before products and then move exponents to coefficients.
2. For condensation, move coefficients to exponents first.
3. Combine added logs by multiplication and subtracted logs by division.
4. Check that the final expression preserves the original domain assumptions.

Common Mistakes to Avoid

A logarithm does not distribute over addition or subtraction inside its argument. Other frequent mistakes include reversing a quotient, losing a negative sign, applying an exponent to only part of a product, and swapping the numerator and denominator in change of base.

Practice Format

Every item includes a transformation, a final equivalent expression, and a direct explanation of why each distractor violates a logarithm rule.