Logarithm Rules Practice Test

Expand and condense logarithmic expressions without confusing products, quotients, powers, and sums.

Logarithm Rules Practice Test

20 logarithm-rule questions with worked transformations and incorrect-choice analysis.

Instant feedback · Worked explanations

Log Rewrite Pressroom

Expand by exposing structure. Condense by rebuilding structure.

Logarithm rules are controlled rewrites of multiplication, division, and powers. Expansion separates a logarithm into simpler pieces; condensation reverses the process. The key is to identify the operation inside the logarithm before applying any rule, move exponents only through the power rule, and never invent a distribution rule for addition or subtraction.

ProductMultiplication inside becomes addition outside.

QuotientDivision inside becomes subtraction outside.

PowerAn exponent inside becomes a coefficient outside.

No sum ruleAddition inside a logarithm does not split.

1. The three legal rewrite rules

These are the structural rules that drive both expansion and condensation.

Product proof

Multiplication becomes addition

\log_{b} (x \cdot y) = \log_{b} (x) + \log_{b} (y)

Use this only when the logarithm argument is a product.

Quotient proof

Division becomes subtraction

\log_{b} (\frac{x}{y}) = \log_{b} (x) - \log_{b} (y)

The numerator log comes first; the denominator log is subtracted.

Power proof

Exponent becomes coefficient

\log_{b} (x^{n}) = n \log_{b} (x)

The exponent multiplies the entire logarithm of the powered base.

2. Expansion works from large structure to small structure

Separate quotients before products, then convert powers and roots into coefficients.

Expansion pressRead the outermost operation first. Do not move exponents until the product and quotient structure is visible.

\log_{b} (\frac{x^{3} \cdot \sqrt{y}}{z^{2}})

1. Split quotient

\log_{b} (x^{3} \cdot \sqrt{y}) - \log_{b} (z^{2})

Top logarithm minus bottom logarithm.

2. Split product

\log_{b} (x^{3}) + \log_{b} (\sqrt{y}) - \log_{b} (z^{2})

The numerator product becomes addition.

3. Move powers

\log_{b} (x^{n}) = n \log_{b} (x)

Use the power rule on every powered factor.

4. Final expanded form

3 \log_{b} (x) + \frac{1}{2} \log_{b} (y) - 2 \log_{b} (z)

No product, quotient, or power remains hidden inside the logarithms.

3. Condensation runs the same rules in reverse

Move coefficients back into exponents before combining sums and differences of logarithms.

Separated form

2 \log_{b} (x) + 3 \log_{b} (y) - \log_{b} (z)

\log_{b} (x^{2}) + \log_{b} (y^{3}) - \log_{b} (z)

First turn coefficients into exponents. This restores the power structure inside each logarithm.

rebuild⇄

Single-log form

\log_{b} (x^{2} \cdot y^{3}) - \log_{b} (z)

\log_{b} (\frac{x^{2} \cdot y^{3}}{z})

Added logarithms become multiplication; the subtracted logarithm becomes division.

4. Red stamp: logarithms do not distribute over addition or subtraction

There is no logarithm rule corresponding to an ordinary sum or difference inside the argument.

Rejected rewrite

\log_{b} (x + y) \neq \log_{b} (x) + \log_{b} (y)

\log_{b} (x - y) \neq \log_{b} (x) - \log_{b} (y)

Product, quotient, and power rules cannot be extended to ordinary addition or subtraction.

Why this matters

When an argument contains an actual sum, keep that sum together unless some other algebraic step—such as factoring—changes the structure into a product. A valid logarithm rule must match the operation that is genuinely present.

5. Roots enter the power rule as fractional exponents

A root is a power, so it can be moved outside a logarithm as a fractional coefficient.

General root

\log_{b} (\sqrt[n]{x}) = \frac{1}{n} \log_{b} (x)

The root index becomes the denominator of the coefficient.

Square root

\log_{b} (\sqrt{x}) = \frac{1}{2} \log_{b} (x)

A square root contributes a one-half coefficient outside the logarithm.

6. The same rules work for common logs, natural logs, and general bases

The algebraic structure does not change just because the logarithm notation changes.

Natural log

Product rule

\ln (x \cdot y) = \ln (x) + \ln (y)

The natural logarithm follows the same multiplication rule.

Common log

Quotient rule

\log (\frac{x}{y}) = \log (x) - \log (y)

Common logarithms obey the same division-to-subtraction rewrite.

General base

Power rule

\log_{b} (x^{n}) = n \log_{b} (x)

The rule remains valid for any permitted logarithm base.

7. Inverse identities collapse logarithm-exponential pairs

A logarithm and exponential expression with matching bases undo each other.

Log after exponent

Return the exponent

\log_{b} (b^{x}) = x

The logarithm asks for the exponent already displayed.

Exponent after log

Return the argument

b^{\log_{b} (x)} = x

The exponential function reverses the logarithm.

Constant rewrite

Turn a constant into a logarithm

4 = \log_{b} (b^{4})

This can make a constant compatible with nearby logarithmic terms during condensation.

8. Constants can be rewritten as logarithms when condensation needs one common language

A numerical constant can be expressed as a logarithm of a matching-base power.

Convert the constant first

4 = \log_{b} (b^{4})

The constant is now expressed as a logarithm with the same base.

Then condense

\log_{b} (x) + 2 = \log_{b} (x) + \log_{b} (b^{2})

\log_{b} (x) + \log_{b} (b^{2}) = \log_{b} (x \cdot b^{2})

Once both terms are logarithms of the same base, addition becomes multiplication inside one logarithm.

9. Change of base is a quotient identity, not a rule to reverse

The logarithm of the argument belongs in the numerator; the logarithm of the original base belongs in the denominator.

Natural-log version

\log_{b} (x) = \frac{\ln (x)}{\ln (b)}

Use this form when natural logarithms are available.

same value⇄

Common-log version

\log_{b} (x) = \frac{\log (x)}{\log (b)}

The new logarithm base may change, but the numerator-over-denominator order does not.

10. Domain assumptions must survive expansion and condensation

Equivalent-looking logarithmic expressions should still be interpreted on the domain where the original logarithms were defined.

Domain watermarkKeep positivity restrictions visible while rewriting; algebraic equivalence does not erase where the original logarithms existed.

Individual logarithms

$x > 0, y > 0, z > 0$

Condensed expression

Do not use the condensed form to silently enlarge the domain beyond the original assumptions.

Quotient structure

Keep the argument of every original logarithm positive before and after the rewrite.

11. Proofing errors: most wrong answers are structurally plausible but illegal

Check whether each rewrite corresponds to a genuine logarithm identity.

Logarithm distributed over addition

$\log_{b} (x + y) \neq \log_{b} (x) + \log_{b} (y)$

Quotient reversed

Numerator log minus denominator log preserves the original fraction order.

Negative sign lost

A denominator becomes a subtraction term during expansion and must stay negative.

Power applied to only part of a product

Rewrite the complete powered factor before moving the exponent outside.

Condensation started before coefficient movement

Coefficients must become exponents before added and subtracted logs are combined.

Change-of-base fraction swapped

Argument on top, original base on the bottom.

Final rewrite audit

Before accepting a transformed logarithmic expression, verify structure, sign, exponent placement, and domain.

For expansion, did I split quotients before products?Read the largest structural operation first.

Did I move powers to coefficients only after separation?Roots count as fractional powers.

For condensation, did I move coefficients back into exponents first?Only then combine sums and differences of logarithms.

Did I convert added logs to multiplication and subtracted logs to division?Keep numerator and denominator order intact.

Did I avoid inventing a sum or difference rule?Logarithms do not distribute over ordinary addition or subtraction.

Did I preserve the original domain assumptions?Every logarithm argument must remain positive on the intended domain.