Quotient Rule of Logarithms Practice Test

Advanced Algebra Practice Test: ACT math skills.

Quotient Rule of Logarithms Practice Test

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Quotient Rule of Logarithms

One ratio. Two log levels.

A quotient inside a logarithm becomes subtraction outside. Move through the gate in either direction while preserving the base, the order of the terms, and every original domain condition.

1Keep one base2Keep numerator first3Check both inputs

\log_{b} (\frac{M}{N}) = \log_{b} M - \log_{b} N

Matching baseBoth logarithms must use the same base.

Valid baseThe base is positive and is not one.

Positive numeratorThe first original argument is positive.

Positive denominatorThe second original argument is positive.

Aqueduct 011Core gate

Division inside becomes subtraction outside

The numerator logarithm stays first. The denominator logarithm is subtracted.

Main rule

Expand a quotient

\log_{b} (\frac{M}{N})

\log_{b} M - \log_{b} N

Condense a difference

\log_{b} M - \log_{b} N

\log_{b} (\frac{M}{N})

Order matters: reversing the two logarithms reverses the quotient. Subtraction is not commutative.

Aqueduct 022Why it works

The quotient rule follows from subtracting exponents

Represent both positive arguments as powers of one valid base.

Structure

Name the logarithm values

\log_{b} M = p and \log_{b} N = q

M = b^{p} and N = b^{q}

Divide powers

Subtract the exponents

\frac{M}{N} = \frac{b^{p}}{b^{q}} = b^{p - q}

\log_{b} (\frac{M}{N}) = p - q

Aqueduct 033Exact values

Evaluate by simplifying the quotient first

When the ratio is a familiar power of the base, the final logarithm is immediate.

Number lock

Base three

\log_{3} 81 - \log_{3} 3

\log_{3} (\frac{81}{3}) = \log_{3} 27 = 3

The separate values are four and one, so their difference is three.

Base ten

\log 1000 - \log 10

\log (\frac{1000}{10}) = \log 100 = 2

An unstated logarithm base is commonly base ten in this school-level setting.

Aqueduct 044Expansion

Separate the numerator and denominator before expanding factors

The entire denominator group is subtracted, so every expanded denominator term receives a minus sign.

Multi-level split

\log (\frac{12 x^{3}}{5 y^{2}})

Split quotient

\log (12 x^{3}) - \log (5 y^{2})

Keep the denominator expression grouped.

Split products

\log 12 + \log x^{3} - \log 5 - \log y^{2}

The minus sign distributes across the denominator logs.

Move powers

\log 12 + 3 \log x - \log 5 - 2 \log y

Exponents become outside coefficients.

Domain

x > 0 and y > 0

The factored expansion assumes positive variable factors.

Aqueduct 055Condensation

Build the numerator from positive terms and the denominator from subtracted terms

Use the power rule before routing factors into the ratio.

Reverse flow

2 \log_{5} x + \log_{5} z - 3 \log_{5} y

Power rule

\log_{5} x^{2} + \log_{5} z - \log_{5} y^{3}

Move each coefficient into its own argument.

Product first

\log_{5} (x^{2} z) - \log_{5} y^{3}

The added terms form the numerator product.

Quotient next

\log_{5} (\frac{x^{2} z}{y^{3}})

The subtracted term becomes the denominator.

Conditions

x > 0, y > 0, z > 0

Every original logarithm requires a positive argument.

Aqueduct 066Equation route

Condense, convert, solve, and test the result

A difference of logarithms can become a rational equation with a clear domain.

Worked solution

\log_{2} (x + 5) - \log_{2} (x - 1) = 2

Domain first

x > 1

Both original arguments must be positive.

Condense

\log_{2} (\frac{x + 5}{x - 1}) = 2

The first argument becomes the numerator.

Convert

\frac{x + 5}{x - 1} = 4

Rewrite the logarithmic equation in exponential form.

Solve

x + 5 = 4 x - 4, x = 3

Clear the denominator and solve the linear equation.

✓

Check:

x = 3

satisfies the original domain, and the argument ratio equals four.

Aqueduct 077Domain screen

A positive quotient does not guarantee two valid original logarithms

Condensation must not enlarge the domain of the expression you started with.

Safety boundary

Domain trap

\log (x - 4) - \log (x + 2)

x > 4

The condensed ratio is also positive below negative two, but both original arguments are then negative. Those values were never allowed.

Reliable habit: write the original positivity conditions before applying any logarithm rule.

Aqueduct 088Sign routing

Plus routes factors upward; minus routes factors downward

Mixed expressions are easier when each complete logarithm is classified by its preceding sign.

Visual organizer

Example routing

\log_{b} A + \log_{b} B - \log_{b} C

\log_{b} (\frac{A B}{C})

The product rule builds the numerator; the quotient rule places the subtracted argument below.

Aqueduct 099Closed gates

Recognize expressions the quotient rule cannot change

The subtraction must occur between complete logarithms with matching bases.

Nonexamples

Subtraction inside one argument

\log_{b} (M - N) \neq \log_{b} M - \log_{b} N

There is no rule that splits an ordinary difference inside a logarithm.

Different bases

\log_{2} 16 - \log_{4} 4

Evaluate separately or change bases first; the quotient rule cannot combine these terms directly.

Recognition check: look for a minus sign between two complete, same-base logarithms.

Aqueduct 1010Study route

Skills Covered and How to Approach

Use a consistent route from rule recognition to final domain verification.

Practice plan

Skills Covered

What this practice set develops.

Recognize the quotient patternFind division inside or subtraction outside.

Expand quotientsDistribute the denominator minus sign correctly.

Condense differencesPreserve numerator-denominator order.

Handle coefficientsUse the power rule before combining.

Solve and screenReject values outside the original domain.

How to Approach

A five-gate routine for test questions.

Identify the requested formDecide whether to expand, condense, evaluate, or solve.

Record the domainRequire every original logarithm argument to be positive.

Match bases and move powersDo not combine until the bases match.

Keep the orderFirst log to numerator, subtracted log to denominator.

Simplify and verifyCheck both the algebra and the original restrictions.

Aqueduct 1111Leak repair

Common Mistakes

Most errors come from reversing the ratio, losing signs, or weakening the domain.

Troubleshooting

Reversing the quotient

The first logarithm supplies the numerator.

Subtracting the arguments

The quotient rule divides arguments; it does not subtract them.

Splitting an inside difference

An ordinary difference inside one log does not separate.

Missing a denominator sign

Every expanded denominator factor contributes a subtracted term.

Combining unlike bases

The bases must match before the quotient rule applies.

Leaving coefficients outside incorrectly

Use the power rule before building the ratio.

Checking only ratio positivity

Each original logarithm argument must be positive separately.

Accepting an excluded solution

Substitute candidates into the original logarithmic equation.

Final aqueduct12Release check

Flow audit complete

A correct quotient-rule solution preserves the base, numerator order, denominator signs, coefficients, and the complete original domain.

Base · Order · Signs · Domain

Do the logarithms have the same valid base?Do not combine different bases directly.

Is the first argument in the numerator?Preserve the subtraction order.

Did every denominator term keep a minus sign?Treat the expanded denominator as a grouped subtraction.

Did each coefficient become the correct exponent?Apply the power rule before condensation.

Does every result satisfy the original domain?Check each logarithm argument separately.

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