Change of Base Formula Practice Test

Advanced Algebra Practice Test: ACT math skills.

Change of Base Formula Practice Test

This test has 20 questions

Instant feedback · Worked explanations

Change of Base Formula

Translate any base. Keep the value.

A logarithm can be rewritten using any other valid base. The original argument becomes the numerator reading, and the original base becomes the denominator reading.

Choose one new base Keep order Divide readings

Original input Place it in the numerator logarithm.

Original base Place it in the denominator logarithm.

New base Use the same new base in both logs.

Final signal Divide before rounding.

The change-of-base formula

Any valid target base produces the same logarithm value.

Core formula

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

Original argument

M > 0

Original base

b > 0 and b \neq 1

New base

k > 0 and k \neq 1

Memory cue: the argument stays on top; the original base goes on the bottom.

Derive the formula from exponential form

The same unknown exponent can be measured using a different logarithm base.

Signal proof

Name the original logarithm

x = \log_{b} M

b^{x} = M

Exponential form says which power of the original base produces the argument.

Take a new-base log

Solve for the exponent

x \log_{k} b = \log_{k} M

x = \frac{\log_{k} M}{\log_{k} b}

Use common logs or natural logs on a calculator

Both choices give the same ratio when the same base is used in numerator and denominator.

Calculator channel

Two equivalent entries

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

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

\log_{2} 7 \approx 2.807

Choose a convenient base for exact evaluation

The new base does not have to be ten or the natural base.

Exact tuning

Use base two

\log_{4} 64

\frac{\log_{2} 64}{\log_{2} 4} = \frac{6}{2} = 3

Both numbers are familiar powers of two.

Common-log confirmation

\log_{5} 125

\frac{\log 125}{\log 5} = 3

The quotient must equal the known exact exponent.

Keep the numerator and denominator in the correct order

The argument is measured above; the original base is measured below.

Wiring order

Original

\log_{b} M

Identify the argument and original base.

Numerator

\log_{k} M

Take the new-base log of the argument.

Denominator

\log_{k} b

Take the same new-base log of the original base.

Ratio

\frac{\log_{k} M}{\log_{k} b}

Divide and round only at the end.

Quick check: reversing the fraction gives the reciprocal, not the original logarithm.

Swapping base and argument creates a reciprocal

This useful identity is a direct consequence of change of base.

Reciprocal link

First direction

\log_{b} a = \frac{\log a}{\log b}

SWAP

Reciprocal direction

\log_{b} a = \frac{1}{\log_{a} b}

✓

Product check:

\log_{b} a \cdot \log_{a} b = 1

Logarithm links can cancel through an intermediate base

Change of base explains a chain identity that often simplifies products.

Bridge identity

\log_{b} a \cdot \log_{a} c = \log_{b} c

Rewrite first

\log_{b} a = \frac{\log a}{\log b}

Use one common measurement base.

Rewrite second

\log_{a} c = \frac{\log c}{\log a}

The intermediate reading appears in both fractions.

Cancel

\frac{\log a}{\log b} \cdot \frac{\log c}{\log a} = \frac{\log c}{\log b}

The remaining ratio is the target logarithm.

Use change of base inside an equation

Translate an unfamiliar logarithm, evaluate it, and solve the remaining exponential statement.

Worked equation

\log_{3} x = \log_{5} 25

Evaluate right

\log_{5} 25 = \frac{\log 25}{\log 5} = 2

Change of base confirms the exact value.

Rewrite equation

\log_{3} x = 2

The unfamiliar right side is now a number.

Convert

x = 3^{2} = 9

Use exponential form and check positivity.

Restrictions protect every part of the fraction

The original logarithm and both new-base logarithms must be defined.

Validity scan

1 Argument positive The original input must be greater than zero.

2 Original base valid It must be positive and not equal to one.

3 New base valid Use one positive base other than one.

4 Denominator nonzero A valid original base makes its new-base log nonzero.

\log_{k} b \neq 0 when b \neq 1

Why: the logarithm of one is zero in every valid base, so an original base of one would create a zero denominator and is already forbidden.

A reliable four-stage conversion workflow

Label the roles before entering numbers into a calculator.

Operating sequence

Conversion sequence

Identify the original argument and original base.

Select one convenient valid new base.

Write the argument log over the original-base log.

Evaluate with full precision and check whether the size is reasonable.

Skills Covered and How to Approach

Separate formula setup, calculator work, exact reasoning, and restriction checks.

Training panel

Skills Covered

Core abilities developed by this practice test.

Identify formula roles Separate argument, original base, and new base.

Use calculator logarithms Convert to common or natural logs accurately.

Choose useful bases Use familiar powers for exact values.

Apply reciprocal identities Recognize base-argument swaps and chains.

Check restrictions Protect arguments, bases, and denominators.

How to Approach

A five-step calculator-safe method.

Read the original logarithm Mark its argument and its base.

Choose one new base Use the same new base in both logs.

Build the fraction Argument on top, original base below.

Keep full precision Round only after the division is complete.

Check the result Compare with nearby integer powers when possible.

Common Mistakes

Most errors are setup errors rather than difficult arithmetic.

Fault monitor

Reversing the fraction

The original argument belongs in the numerator.

Putting the new base below

The denominator contains the log of the original base.

Using two different new bases

Numerator and denominator must use the same one.

Rounding each logarithm early

Keep calculator precision until the final quotient.

Forgetting parentheses

Enter the complete numerator and denominator carefully.

Allowing an invalid original base

A logarithm base must be positive and not one.

Allowing a nonpositive argument

Real logarithm inputs must be positive.

Assuming only base ten works

Any valid new base gives the same ratio.

Final conversion audit

Inspect order, base consistency, restrictions, precision, and reasonableness.

Output approved

Signal conversion locked

A correct change-of-base setup preserves the original logarithm value while expressing it as one carefully ordered ratio.

Input · Base · Ratio · Check

Is the original argument positive? The numerator logarithm must be defined.

Are both bases valid? Each must be positive and different from one.

Is the argument log in the numerator? Keep the original base log in the denominator.

Did both logs use the same new base? Mixing bases destroys the formula.

Was rounding delayed until the end? Confirm the result against nearby powers.

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