Logarithm Properties Practice Test

Advanced Algebra Practice Test: ACT math skills.

Logarithm Properties Practice Test

This test has 20 questions

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Logarithm Properties

Weave terms. Preserve meaning.

Logarithm properties translate multiplication, division, and powers into addition, subtraction, and coefficients. The patterns are compact, but they work only with matching valid bases and positive arguments.

Spool 01
Conditions

Every property begins with valid inputs

Restrictions are part of the identity, not a separate afterthought.

Positive base

b > 0

A real logarithm uses a positive base.

Excluded base

b \neq 1

A base of one has no logarithmic inverse.

Positive argument

M > 0

Every argument in the original expression must be positive.

When several logs appear: write a positivity condition for each original argument before expanding, condensing, or solving.

Spool 02
Core rules

The three main properties move operations across the logarithm

Multiplication, division, and exponents correspond to three different outside operations.

Product property

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

A product inside becomes addition outside.

Quotient property

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

A quotient inside becomes subtraction outside.

Power property

\log_{b} (M^{k}) = k \log_{b} M

An exponent inside becomes a multiplier outside.

Direction works both ways: expanding separates one logarithm; condensing combines several same-base logarithms into one.

Spool 03
Product

Add logarithms only when their arguments multiply

The property does not distribute over ordinary addition inside an argument.

Correct numerical use

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

\log_{2} 32 = 5

The arguments multiply because the logs are added.

Common false rule

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

There is no sum property for an addition already inside one logarithm.

Spool 04
Quotient

Subtraction outside records division inside

The order of the logarithms determines the order of the quotient.

Start

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

Both logarithms use base three.

Condense

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

The first argument becomes the numerator.

Simplify

\log_{3} 27 = 3

The quotient is a familiar power of three.

Order check: reversing the two logarithms takes the reciprocal and changes the sign of the final value.

Spool 05
Power

The power property moves an exponent, not a base

The multiplier applies to the complete logarithm after expansion.

Move outward

\log_{5} (2^{3}) = 3 \log_{5} 2

The exponent becomes a coefficient.

Move inward

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

A one-half multiplier becomes a square root.

Parentheses matter: a multiplier outside a sum of logarithms must distribute to every term before each coefficient can become an exponent.

Spool 06
Anchor facts

Identity and inverse properties shorten expressions

These facts come directly from the inverse relationship between logarithms and exponentials.

Logarithm of one

\log_{b} 1 = 0

A valid base to the zero power equals one.

Logarithm of the base

\log_{b} b = 1

A valid base to the first power equals itself.

Inverse cancellation

b^{\log_{b} M} = M

Exponentiation and the matching logarithm undo each other.

Spool 07
Expansion

Expand from outer operations to inner factors

Separate products and quotients before moving powers outward.

\log_{2} (\frac{8 x}{y})

Quotient

\log_{2} (8 x) - \log_{2} y

Division becomes subtraction.

Product

\log_{2} 8 + \log_{2} x - \log_{2} y

Multiplication becomes addition.

Evaluate

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

The numerical logarithm equals three.

x > 0 and y > 0

Spool 08
Condensing

Condense by reversing the properties in a controlled order

Move coefficients inward first, then combine addition and subtraction.

2 \log_{5} x - \log_{5} y + \frac{1}{2} \log_{5} z

Powers

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

Coefficients become exponents or roots.

Product

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

The added logarithms combine first.

Quotient

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

The subtraction places the second argument below.

x > 0, y > 0, z > 0

Spool 09
Domain-aware power

An even power can hide a sign condition

Expanding an even-powered argument requires attention to absolute value.

Original squared argument

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

x \neq 0

Positive and negative nonzero inputs both produce a positive square.

Preserve domain

Domain-safe expansion

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

Writing only $2 \log_{b} x$ would incorrectly exclude negative inputs.

School-level safe habit: when variables are stated positive, use the ordinary power rule directly. Otherwise compare the domains before claiming two forms are equivalent.

Spool 10
Equation use

Properties can reduce several logarithms to one equation

Condense, convert, solve, and filter every algebraic candidate.

\log x + \log (x - 3) = 1

Domain

x > 3

Both common-log arguments must be positive.

Condense

x (x - 3) = 10

A common logarithm equal to one has argument ten.

Factor

(x - 5) (x + 2) = 0

The candidates are five and negative two.

Solution

x = 5

Only five belongs to the original domain.

Spool 11
Change of base

Change of base rewrites an unfamiliar base without changing the value

The new logarithms in the numerator and denominator must use the same base.

\log_{b} M = \frac{\log_{c} M}{\log_{c} b} = \frac{\ln M}{\ln b}

Calculator form

\log_{2} 7 = \frac{\ln 7}{\ln 2}

Reasonableness check

2 < \log_{2} 7 < 3

Seven lies between the second and third powers of two.

Precision habit: keep the full calculator value during later work and round only as the question requests.

Spool 12
Process map

Expansion and condensation run in opposite directions

Use the direction that makes the target expression simpler.

Direction guide

Expand when factors, quotients, or powers need to become separate terms.

Condense when several same-base terms need to become one logarithm.

Move powers only after the product and quotient structure is clear.

Track all original domain restrictions in either direction.

Spool 13
Study plan

Skills Covered and How to Approach

Separate pattern recognition, domain reasoning, and algebraic simplification.

Skills Covered

Core skills measured by the practice test.

Recognize the propertyMatch products, quotients, powers, and inverse forms.

Expand logarithmsSeparate factors in a controlled order.

Condense logarithmsMove coefficients inward and combine matching bases.

Track the domainPreserve positivity conditions through every rewrite.

Use properties in equationsCondense before solving and reject invalid candidates.

How to Approach

A five-step property routine.

Read the target form

Decide whether the task asks for expansion, condensation, evaluation, or solving.

Check bases and arguments

Require matching bases and record positive inputs.

Identify inside operations

Products, quotients, and powers determine the property.

Rewrite one layer at a time

Preserve order, signs, coefficients, and parentheses.

Check equivalence

Compare domains and, when useful, test simple numerical inputs.

Spool 14
Tangled threads

Common Mistakes

Most property errors come from inventing a rule, reversing an order, or dropping a restriction.

Inventing a sum property

A logarithm of a sum does not split into two logarithms.

Combining different bases

Product and quotient properties require the same base.

Reversing a quotient

The first logarithm supplies the numerator.

Moving the base as a coefficient

The power rule moves an exponent, not the logarithm base.

Forgetting to distribute

An outside multiplier on a group applies to every logarithm in it.

Losing a square-root exponent

A square root corresponds to an exponent of one-half.

Ignoring absolute value

An even-powered argument may allow negative nonzero inputs.

Dropping original restrictions

A condensed expression does not erase earlier positivity conditions.

Final property audit

Check each thread before accepting the rewritten expression.

Are the bases valid and identical where required?Do not combine unrelated logarithm bases.

Are all original arguments positive?Keep the domain beside every transformation.

Does each outside operation match the inside operation?Addition means product; subtraction means quotient.

Did every coefficient become the correct exponent?Fractions may produce roots or reciprocal powers.

Are the original and rewritten forms equivalent?Check both value and domain.

Pattern secured

A property rewrite is complete only when its operation, base, coefficient, and domain all remain consistent.

Match · Expand · Condense · Check

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