Exponential Equations with Different Bases Practice Test

Advanced Algebra Practice Test: ACT math skills.

Exponential Equations with Different Bases Practice Test

This test has 20 questions

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After-test translation guide

Translate different bases into logarithmic language

When exponential bases are genuinely unrelated, their exponents cannot be compared directly. Logarithms translate both sides into products, allowing an ordinary linear equation to emerge.

Inspect bases→Take logs→Bring exponents forward

Logarithm bridge

Gate 1Are the bases secretly related?

Gate 2Are both sides positive?

Gate 3Were logs applied to both sides?

Gate 4Was the decimal verified?

Method choice

Different-looking is not always different-base

Test for a shared power family before using logarithms. Exact rewriting is shorter whenever it is available.

Related bases

Rewrite both sides with one smaller base and equate the exponent expressions.

8^{u} = 4^{v} \Rightarrow 2^{3 u} = 2^{2 v}

Unrelated bases

Take the same logarithm of both positive sides and bring each exponent forward.

a^{u} = b^{v} \Rightarrow u \ln (a) = v \ln (b)

Isolated target

If one power equals a positive constant, take logs or use change of base.

a^{u} = k \Rightarrow u = \frac{\ln (k)}{\ln (a)}

Never equate exponents merely because both sides are exponential. The one-to-one property requires the same base, not simply two positive bases.

Why the logarithm bridge works

The power rule changes a logarithm of a power into the exponent multiplied by the logarithm of its base. This creates an equation that can usually be solved with linear algebra.

Common logarithm

Use the calculator key labeled log on both sides.

u \log (a) = v \log (b)

Natural logarithm

Using ln produces the same solution because both sides use the same logarithm type.

u \ln (a) = v \ln (b)

Crossing 01

Both exponents contain the variable

Take logs, expand only the exponent factors, and collect the variable terms.

Two-base equation

Start with unrelated bases

2^{x + 1} = 7^{x}

Take natural logs

(x + 1) \ln (2) = x \ln (7)

Collect the variable

\ln (2) = x [\ln (7) - \ln (2)]

Divide and approximate

x = \frac{\ln (2)}{\ln (7) - \ln (2)} \approx 0.553

Translated result:

x \approx 0.553

. Substitution should make both positive powers approximately equal.

Crossing 02

Linear expressions appear in both exponents

Preserve parentheses when applying the logarithm power rule.

Full exponents

Read both exponent expressions

3^{2 x - 1} = 5^{x + 2}

Bring both exponents forward

(2 x - 1) \ln (3) = (x + 2) \ln (5)

Group variable and constant terms

x [2 \ln (3) - \ln (5)] = \ln (3) + 2 \ln (5)

Calculate at the end

x \approx 7.345

Translated result:

x \approx 7.345

. The exact logarithmic expression is preferable until the final calculator step.

Trap: equate exponents

The bases 3 and 5 are not identical.

Trap: lose parentheses

Each full exponent multiplies its logarithm.

Trap: decimal too soon

Rounded log values can shift a close answer choice.

Crossing 03

Separate exponential factors before taking logs

A ratio of bases can simplify an equation with matching variable exponents.

Ratio method

Start with coefficients and powers

4 3^{x} = 5 2^{x}

Divide the exponential factors

\frac{3^{x}}{2^{x}} = \frac{5}{4}

Combine the quotient of powers

{(\frac{3}{2})}^{x} = \frac{5}{4}

Use change of base

x = \frac{\ln (\frac{5}{4})}{\ln (\frac{3}{2})} \approx 0.550

Translated result:

x \approx 0.550

. Dividing first produces one clean exponential expression.

Crossing 04

Do not use logs when an exact common base is available

Different visible bases can still reduce to one shared base.

Exact shortcut

Recognize a shared power family

8^{x - 2} = 4^{2 x + 1}

Rewrite with base 2

2^{3 x - 6} = 2^{4 x + 2}

Equate exponents

3 x - 6 = 4 x + 2

Solve exactly

x = - 8

Exact result:

x = - 8

. The common-base route avoids calculator approximation entirely.

Use matching log keys

Common log and natural log both work, but do not mix types within the same derived equation.

\frac{\ln (k)}{\ln (a)} = \frac{\log (k)}{\log (a)}

Group the denominator

Enter every sum or difference in a denominator inside parentheses.

\frac{\ln (2)}{[\ln (7) - \ln (2)]}

Round only once

Keep the exact logarithmic expression through the algebra and approximate only the final requested value.

Different-base decision ledger

Choose exact conversion before approximation

Equation pattern	Best first move	Resulting structure	Main caution
Bases belong to one power family	Rewrite with a common base.	Linear equation between exponents.	Multiply conversion exponents through parentheses.
Unrelated positive bases on both sides	Take the same logarithm of both sides.	Exponent times logarithm of base.	Do not equate original exponents.
Same variable exponent with coefficients	Divide to form a ratio of powers.	One exponential equation with a fractional base.	Divide coefficients in the correct order.
One isolated power equals a positive target	Use logarithms or change of base.	Exact logarithmic quotient.	The target must be positive.
Decimal answer requested	Calculate after all symbolic algebra.	Rounded approximation.	Use parentheses around grouped numerator and denominator.

Verify on both sides of the bridge

An approximate solution should make the two original exponential expressions nearly equal. The comparison also catches a reversed logarithm difference or a missing exponent factor.

Return to the original equation with its original bases.
Substitute the unrounded value when possible.
Evaluate the left and right sides separately.
Check that both outputs are positive and close at the requested precision.
If an exact common-base route existed, prefer its exact answer.
State whether the reported result is exact or approximate.

Skills Covered

Distinguishing genuinely different bases from related power families.
Taking common or natural logarithms of both sides.
Using the logarithm power rule to bring exponents forward.
Solving the resulting linear equation in the variable.
Combining powers with the same exponent into a base ratio.
Entering grouped logarithmic expressions accurately.
Checking exact and approximate answers in the original equation.

Common Mistakes

Equating exponents while the bases remain different.
Taking a logarithm on only one side.
Dropping parentheses around a multi-term exponent.
Using the logarithm power rule on only part of an exponent.
Reversing a difference of logarithms when collecting terms.
Entering a quotient without grouping its denominator.
Using decimals when an exact common-base solution is available.

Final translation audit

Before accepting an answer, confirm that the equation truly required a logarithmic translation and that both original exponential sides agree after substitution.

InspectAre the bases related or genuinely different?

TranslateWas the same logarithm applied to both sides?

SolveWere complete exponent factors collected correctly?

VerifyDo the original outputs agree at the stated precision?

Practice note: keep logarithms symbolic until the final calculator step. The examples in this review block are illustrative and are not copies of the test questions.