Solving Logarithmic Equations Practice Test

Advanced Algebra Practice Test: ACT math skills.

Solving Logarithmic Equations Practice Test

This test has 20 questions

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Solving Logarithmic Equations

Route the structure. Screen every root.

Logarithmic equations do not all use the same first move. A single logarithm usually converts to exponential form, equal same-base logarithms allow equal arguments, and several logarithms often need to be condensed. Every route still ends at the original domain.

Dispatch ruleWrite restrictions first. Then choose the transformation that reduces the number of logarithms without changing their bases or losing parentheses.

RestrictFind the intersection of all argument conditions.

RecognizeCount logs and compare their bases.

TransformConvert, match, or condense.

SolveKeep every algebraic candidate.

ScreenReturn to the original equation.

Yard 01
Domain manifest

Restrictions are recorded before any transformation

Condensing can hide separate restrictions, so keep the original domain beside the algebra.

Restriction example

\log_{3} (x - 2) + \log_{3} (x + 4)

x > 2

The first argument requires a value greater than two; that stricter condition also makes the second argument positive.

Yard 02
Direct line

One isolated logarithm converts directly

Keep the complete argument together when it becomes the exponential output.

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

Restriction

x > \frac{1}{2}

The original argument must be positive.

Convert

2 x - 1 = 3^{2}

The entire argument equals nine.

Solve

x = 5

The candidate belongs to the domain.

Verify

\log_{3} 9 = 2

The original equation is true.

Yard 03
Matching line

Equal logarithms with the same base have equal arguments

The shortcut works only where both original logarithms are defined.

\log_{5} (x + 6) = \log_{5} (3 x - 2)

Match and solve

x + 6 = 3 x - 2

x = 4

Check both arguments

4 + 6 = 10

3 (4) - 2 = 10

Both arguments are positive and equal.

Yard 04
Product line

A sum of logarithms condenses to a product

The resulting quadratic may create candidates that the domain rejects.

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

Domain

x > 1

Both original arguments must be positive.

Condense

(x - 1) (x + 1) = 2^{3}

The sum becomes a product.

Quadratic

x^{2} = 9

The candidates are positive three and negative three.

✓

Accepted candidate

$x = 3$ satisfies the domain and the original equation.

Rejected candidate

$x = - 3$ makes the first argument negative.

Yard 05
Quotient line

A difference of logarithms condenses to a quotient

Keep restrictions from the separate original arguments.

\log_{3} (x + 5) - \log_{3} (x - 1) = 1

Condense and convert

\frac{x + 5}{x - 1} = 3^{1}

The original restrictions combine to $x > 1$ .

Solve and verify

x + 5 = 3 x - 3

x = 4

The candidate is greater than one and passes.

Yard 06
Power junction

An outside multiplier and a power inside are not identical domains

Both may involve the power property, but the original argument still controls the solution set.

Outside multiplier

2 \log_{5} x = 4

\log_{5} x = 2

x = 25

The original argument requires a positive value.

Compare domains

Power inside the argument

\log_{5} (x^{2}) = 4

x^{2} = 625

x = - 25 or x = 25

Both nonzero values make the squared argument positive.

Yard 07
Root screening

A condensed common-log equation can produce an invalid root

Factor the quadratic, then compare every candidate with the original domain.

\log x + \log (x - 6) = \log 7

Domain

x > 6

Both common-log arguments must be positive.

Condense

x (x - 6) = 7

Equal logs give equal positive arguments.

Factor

(x - 7) (x + 1) = 0

The candidates are seven and negative one.

Solution

x = 7

Only seven lies in the original domain.

Yard 08
Base transfer

Different bases must be evaluated or rewritten before comparison

Do not set arguments equal when the logarithm bases differ.

Evaluate the known side

\log_{3} (2 x + 1) = \log_{9} 81

\log_{9} 81 = 2

Finish with the remaining base

2 x + 1 = 3^{2}

x = 4

When a base is unfamiliar: change of base rewrites it as

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

. Keep full calculator precision until the final answer.

Yard 09
Closed track

Conflicting restrictions prove that no real solution exists

A domain intersection can close the route before any equation solving.

\log_{2} (x - 4) = \log_{2} (2 - x)

First track condition

x > 4

Second track condition

x < 2

No real solution

Reason: no real number satisfies both original restrictions, so the two logarithms are never defined together.

Yard 10
Final inspection

Verification is part of solving, not an optional extra

Algebra produces candidates; the original logarithmic equation approves solutions.

Inspection sequence

Check every original argument, not only a condensed product or quotient.

Confirm that each property used matching bases.

Retain all algebraic roots until screening.

Substitute each surviving candidate into the original equation.

Definition checkEvery logarithm argument is strictly positive.

Structure checkThe conversion or property matches the original form.

Equality checkBoth original sides are defined and equal.

Yard 11
Study board

Skills Covered and How to Approach

Use a consistent solving routine while allowing the first transformation to change with the structure.

Skills Covered

Core reasoning measured by the practice test.

Write logarithm domainsIntersect all argument and base restrictions.

Recognize equation structureDistinguish isolated, equal, and multiple logarithms.

Use logarithm propertiesCondense sums, differences, and powers correctly.

Solve resulting algebraHandle linear, quadratic, and exponential forms.

Screen candidatesReject extraneous values and detect no-solution cases.

How to Approach

A five-step dispatch routine.

Record the original domain

Require each argument to be positive before condensing.

Select the shortest valid route

Convert one log, match equal logs, or condense several logs.

Transform carefully

Match bases, preserve parentheses, and keep complete arguments.

Solve without early rejection

Find every algebraic candidate and keep exact values.

Screen in the original equation

Check definition, equality, and the requested answer form.

Yard 12
Wrong turns

Common Mistakes

Most errors come from choosing the wrong route or forgetting the original domain.

Skipping restrictions

A transformed equation may accept values forbidden by the original logs.

Combining different bases

Product and quotient properties require matching bases.

Adding arguments inside

A sum of logarithms becomes a logarithm of a product.

Losing quotient restrictions

Each original argument must be positive separately.

Confusing a multiplier with an inside power

The two original equations may have different domains.

Discarding a quadratic root too early

Find all candidates before applying the domain.

Keeping every algebraic root

A candidate is not a solution until it passes the original equation.

Rounding during change of base

Keep calculator precision until the final requested value.

Final yard
Release audit

Solution release checklist

Release only answers that complete the algebra and survive the original restrictions.

Cleared for departure

A logarithmic solution is valid only after the route, domain, and original equality have all been checked.

Restrict · Route · Solve · Screen

Did I record every original restriction?Each argument must remain strictly positive.

Did I choose the route from the structure?Convert, match, or condense as appropriate.

Were all properties used with matching bases?Preserve the correct product, quotient, or power.

Did I solve for every algebraic candidate?Keep exact values until screening is complete.

Does each reported value satisfy the original equation?Confirm both definition and equality.

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