Logarithmic Equation Word Problems Practice Test

Advanced Algebra Practice Test: ACT math skills.

Logarithmic Equation Word Problems Practice Test

This test has 20 questions

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After-test field guide

Turn the story into a logarithmic equation

Word problems rarely announce which logarithm to use. They describe a starting amount, a repeated factor, a measured scale, or a target value. Your job is to identify the model, isolate the exponent, calculate carefully, and translate the number back into the situation.

Observe→Model→Isolate→Interpret

StartWhat is known?

ChangeWhat repeats?

TargetWhat must be reached?

MeaningWhat unit belongs?

Model cardRecognize the structure before touching the calculator.

When does a word problem need a logarithm?

A logarithm is especially useful when the unknown appears in an exponent or when the situation already uses a logarithmic scale. First decide which of those structures is present.

Unknown time or count

A quantity repeatedly grows or decays, and the problem asks when it reaches a target.

y = a b^{t}

Logarithmic measurement

The stated scale compresses a large range, so equal scale increases represent multiplicative changes.

S = k \log_{10} (R)

Known base, unknown exponent

Rewrite the exponential statement and solve the exponent by taking a logarithm of both sides.

t = \frac{\log (y / a)}{\log (b)}

Logarithms do not create the model. The words determine the model; logarithms are the tool that isolates the exponent or reverses a logarithmic scale.

Report 01

Savings growth: solve for time

A deposit of $2,000 grows at 6% per year, compounded annually. When will it first exceed $3,000?

Growth model

Translate the story

The initial amount is $2,000, the annual growth factor is $1 + 0.06 = 1.06$ , and the target is $3,000.

3000 = 2000 {(1.06)}^{t}

Isolate the exponential part

Divide by the starting amount before taking logarithms.

1.5 = {(1.06)}^{t}

Solve the exponent

Either common logarithms or natural logarithms give the same quotient.

t = \frac{\ln (1.5)}{\ln (1.06)} \approx 6.96

Interpret the timing

The calculation gives a crossing time just before the seventh annual compounding point. Because the balance changes once per year in this model, the first whole year that exceeds the target is year 7.

Context answer: The deposit first exceeds $3,000 after 7 annual compounding periods. A decimal time and a whole-period answer are not automatically interchangeable.

Trap: add 6%

Repeated percentage growth multiplies by the same factor; it does not add the same dollar amount.

Trap: round down

At 6 years the account has not yet crossed the target, so 6 is too early.

Trap: omit units

The unknown measures years, not dollars or percent.

Report 02

Medicine decay: use a half-life model

A medicine has a half-life of 4 hours. How long does it take for 80 milligrams to decrease to 10 milligrams?

Decay model

Keep time and half-life in the same unit

Both are measured in hours, so the exponent counts how many 4-hour half-lives have passed.

10 = 80 {(\frac{1}{2})}^{\frac{t}{4}}

Divide by the initial amount

The remaining fraction is one eighth.

\frac{1}{8} = {(\frac{1}{2})}^{\frac{t}{4}}

Use structure before a calculator

Since $\frac{1}{8} = {(\frac{1}{2})}^{3}$ , the exponent must equal 3.

\frac{t}{4} = 3

Recover the requested time

Multiply the number of half-lives by 4 hours per half-life.

t = 12

Context answer: It takes 12 hours. A logarithmic method would also work, but recognizing an exact power is faster and reduces calculator error.

Read logarithmic scales as ratios, not ordinary differences

On a base-10 logarithmic scale, an increase of one scale unit often represents a tenfold change in the underlying quantity. The exact multiplier depends on the model's coefficient, so always read the formula given in the problem.

Earthquake-style model

If magnitude is based on amplitude ratio, a difference of 2 corresponds to a factor of 100.

M_{2} - M_{1} = \log_{10} (\frac{A_{2}}{A_{1}})

Sound intensity level

A 20-decibel increase corresponds to an intensity ratio of 100 because the formula includes a factor of 10.

Δ L = 10 \log_{10} (\frac{I_{2}}{I_{1}})

Report 03

Sound level: reverse a logarithmic formula

One sound is 30 decibels louder than a reference sound. How many times as intense is it?

Scale model

Use the level-difference model

Let the intensity ratio be $R$ . The problem gives the level difference.

30 = 10 \log_{10} (R)

Isolate the logarithm

Divide both sides by 10.

3 = \log_{10} (R)

Convert to exponential form

The logarithm states the exponent on base 10.

R = 10^{3} = 1000

State a ratio answer

The result compares intensities. It is not an additional number of decibels.

Context answer: The sound is 1,000 times as intense as the reference sound.

Trap: answer 30

Thirty is the level difference already supplied, not the requested intensity ratio.

Trap: use a factor of 10

The coefficient outside the logarithm must be removed before converting forms.

Trap: treat as linear

A decibel increase represents multiplication of intensity, not simple addition.

Report 04

Acidity: solve a negative logarithm

A solution has a pH of 3. Find its hydrogen-ion concentration in moles per liter.

Inverse scale

Substitute into the model

The concentration is the positive quantity inside the logarithm.

3 = - \log_{10} (C)

Handle the negative sign first

Multiply both sides by negative one.

- 3 = \log_{10} (C)

Convert to exponential form

The concentration must be positive, as required by the logarithm's domain.

C = 10^{- 3} = 0.001

Attach the given unit

The numerical answer alone is incomplete because the question asks for a concentration.

Context answer: The hydrogen-ion concentration is 0.001 mole per liter.

Language-to-model translation ledger

Use the clue, then verify the units

Wording clue	Likely structure	What to identify	Frequent error
Grows by the same percent each period	$A = P {(1 + r)}^{t}$	Initial amount, decimal rate, number of periods, target.	Using the percent as the growth factor instead of adding one.
Decreases by the same percent each period	$A = P {(1 - r)}^{t}$	Remaining factor and compatible time units.	Using a growth factor greater than one.
Half-life is given	$Q = Q_{0} {(\frac{1}{2})}^{\frac{t}{h}}$	Half-life $h$ , elapsed time $t$ , and remaining amount.	Writing the exponent as a product instead of a number of half-lives.
Scale difference or level is given	$S = k \log (R)$	Scale coefficient, logarithm base, requested ratio.	Assuming every one-unit increase means exactly the same multiplier.
How long until a target?	$t = \frac{\log (target / initial)}{\log (factor)}$	Whether time is continuous or counted in whole periods.	Rounding before deciding what the story requires.

A dependable six-step workflow

Use this route when a word problem contains more information than you need. It keeps the model, algebra, units, and final sentence connected.

Underline the initial value, target value, repeating factor or scale rule, and requested unit.
Define the unknown with its unit before writing an equation.
Write the exponential or logarithmic model and substitute only matching units.
Isolate the exponential expression or logarithm before using a calculator.
Keep several decimal places during the calculation; round only the final result.
Check the result in the original model and write one sentence that answers the story.

ReasonablenessA calculator result is not yet a complete answer.

Check direction, domain, unit, and size

Direction check

Growth should move upward toward a larger target. Decay should move downward toward a smaller target. A negative time often signals a reversed ratio or incorrect factor.

Domain check

Every logarithm argument must be positive. Initial quantities, target ratios, concentrations, and intensities used inside logarithms must satisfy that condition.

argument > 0

Unit and size check

Compare with nearby powers or periods. If five doublings are too small and six are large enough, the time should lie between those two milestones.

Skills Covered

Recognizing exponential growth, exponential decay, doubling, and half-life models.
Reversing base-10 logarithmic measurement formulas.
Using common or natural logarithms to isolate an exponent.
Converting rates to growth or decay factors.
Keeping time periods and measurement units consistent.
Interpreting decimal results and whole-period thresholds.
Checking logarithm domains and verifying answers in context.

Common Mistakes

Choosing a linear model for repeated percentage change.
Using a percent such as 6 instead of its decimal form or correct factor.
Taking a logarithm before isolating the exponential expression.
Dropping a coefficient or negative sign outside a logarithm.
Mixing months, years, hours, or other periods in one exponent.
Rounding during intermediate steps and shifting a threshold answer.
Reporting a bare number without the requested unit or interpretation.

Final field audit

Before accepting an answer, confirm that the equation matches the story and that the final statement answers the actual question rather than merely reporting a calculator display.

ModelDoes the repeated change or scale rule match the wording?

AlgebraWas the logarithm used after the correct expression was isolated?

ValidityAre logarithm arguments positive and time values meaningful?

ContextDoes the final number include units and sensible rounding?

Practice note: sketch the quantities and write the model before reaching for a calculator. The examples in this review block are illustrative and are not copies of the test questions.