Product Rule of Logarithms Practice Test

Advanced Algebra Practice Test: ACT math skills.

Product Rule of Logarithms Practice Test

This test has 20 questions

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Product Rule of Logarithms

Two log streams. One product channel.

When logarithms have the same valid base, addition outside corresponds to multiplication inside. The product rule can evaluate expressions, expand products, condense sums, and turn multi-log equations into ordinary algebra.

Match basesCheck inputsMultiply inside

Foundry 01

The product rule

The central merge pattern.

Core identity

Addition outside means multiplication inside

The rule works in either direction when the bases match and the original arguments are positive.

\log_{b} M + \log_{b} N = \log_{b} (M N)

Base condition

b > 0 and b \neq 1

First argument

M > 0

Second argument

N > 0

Important: the positivity requirements come from the two original logarithms and remain in force after condensation.

Foundry 02

Why it works

Exponent laws power the rule.

Structural proof

The sum of exponents creates a product

Write each positive argument as a power of the same base.

Name the logarithm values

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

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

Multiply

Use the exponent rule

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

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

Foundry 03

Numerical merges

Combine first, evaluate second.

Exact values

Familiar powers make the product rule visible

Multiplying the arguments should lead to the same result as adding the separate logarithm values.

Base two

\log_{2} 4 + \log_{2} 8

\log_{2} 32 = 5

The separate values are two and three.

Base three

\log_{3} 9 + \log_{3} 27

\log_{3} 243 = 5

The separate values are two and three again.

Foundry 04

Expand a product

Split one log into a sum.

Reverse conveyor

Each positive factor becomes its own logarithm

Expansion is useful when factors are easier to evaluate or simplify separately.

\log_{5} (25 x y)

Split factors

\log_{5} 25 + \log_{5} x + \log_{5} y

Three positive factors create three terms.

Evaluate

2 + \log_{5} x + \log_{5} y

The numerical logarithm equals two.

Conditions

x > 0 and y > 0

Both variable factors are stated positive.

Foundry 05

Condense a sum

Merge matching logarithms.

Forward conveyor

Multiply arguments without adding them

Condensation creates one logarithm whose argument is the product of the originals.

Variable and constant

\log_{5} x + \log_{5} 4

\log_{5} (4 x)

Three factors

\log_{b} x + \log_{b} y + \log_{b} z

\log_{b} (x y z)

Do not write a sum inside: added logarithms correspond to multiplied arguments, not added arguments.

Foundry 06

Coefficients first

Power rule before product rule.

Two-step merge

Move outside coefficients into exponents before multiplying

Each coefficient belongs to its own logarithm.

2 \log_{3} x + \log_{3} y

Power rule

\log_{3} x^{2} + \log_{3} y

The coefficient becomes an exponent.

Product rule

\log_{3} (x^{2} y)

The two arguments now multiply.

Conditions

x > 0 and y > 0

The original logarithms require positive inputs.

Fractional coefficient

\frac{1}{2} \log_{b} M + \log_{b} N

Condensed result

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

Foundry 07

Solve with a merge

One log leads to algebra.

Candidate casting

The product rule can reveal a quadratic equation

Write the domain first, condense, solve, and screen both roots.

\log_{2} (x - 1) + \log_{2} (x - 3) = 3

Domain

x > 3

Both original arguments must be positive.

Condense

(x - 1) (x - 3) = 8

The one logarithm converts to exponential form.

Factor

(x - 5) (x + 1) = 0

The candidates are five and negative one.

✓

Accepted

$x = 5$ makes both arguments positive.

Rejected

$x = - 1$ violates the original domain.

Foundry 08

Domain trap

A product can hide invalid inputs.

Safety inspection

A positive product is not enough for the original two logs

Each separate argument must be positive, even if their product is positive elsewhere.

Example structure

\log (x - 5) + \log (x + 1)

x > 5

The condensed product is also positive for values below negative one, but those values make both original arguments negative and are forbidden.

Keep the original domain: condensation preserves values only on the domain shared by the original logarithms.

Foundry 09

Nonexamples

Know when not to merge.

Fault detection

The rule does not apply to every visible plus sign

Check both the location of addition and the logarithm bases.

Addition inside one argument

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

There is no logarithm sum rule.

Different bases

\log_{2} 8 + \log_{4} 16

The two values can be evaluated separately, but the product rule cannot combine them directly.

Fast recognition: the valid pattern has a plus sign between complete logarithms that share one base.

Foundry 10

Merge workflow

A repeatable quality check.

Production line

Four inspections prevent most product-rule errors

Read the original expression before multiplying any arguments.

Production sequence

Confirm that every logarithm uses the same valid base.

Record positivity conditions for each original argument.

Use the power rule on any outside coefficient.

Multiply the resulting arguments inside one logarithm.

Foundry 11

Study board

Skills and test approach.

Training shift

Skills Covered and How to Approach

Separate rule recognition, coefficient handling, domain reasoning, and final simplification.

Skills Covered

Core abilities measured by the practice test.

Recognize the patternFind addition between same-base logarithms.

Expand productsTurn positive factors into separate logarithms.

Condense sumsMultiply arguments inside one logarithm.

Handle coefficientsUse the power rule before merging.

Preserve domainsScreen expressions and equation solutions.

How to Approach

A five-step foundry routine.

Identify the target

Decide whether to expand, condense, evaluate, or solve.

Check bases and arguments

Match bases and write all original positivity conditions.

Move coefficients inward

Convert each outside multiplier into an exponent.

Multiply the factors

Place every argument inside one logarithm.

Simplify and verify

Check the requested form and preserve the original domain.

Foundry 12

Common Mistakes

Faults in the merge line.

Repair manual

Adding the arguments

The product rule multiplies arguments.

Splitting a sum inside

A logarithm of an ordinary sum does not expand.

Combining different bases

The logarithms must share one valid base.

Ignoring an outside coefficient

Move it into an exponent before merging.

Dropping a factor

Every added logarithm contributes one factor.

Keeping only product positivity

Each original argument must be positive separately.

Accepting every quadratic root

Equation candidates must satisfy the original domain.

Expanding when condensation is requested

Read the required final form before choosing a direction.

Final foundry

Release audit

Inspect the completed merge.

Approved output

Product channel cleared

A correct condensation preserves matching bases, every factor, all coefficients, and the full original domain.

Match · Power · Multiply · Verify

Do all logarithms have the same valid base?Different bases cannot merge directly.

Is every original argument positive?Keep each separate restriction after condensation.

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

Did every added logarithm contribute a factor?Multiply rather than add the arguments.

Does the result match the requested form?Check expansion, condensation, evaluation, or solution.

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