Change of Base Formula Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Change of Base Formula.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A formula for converting a logarithm from one base to another: $\log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}$ .

Your calculator only has $\ln$ and $\log_{10}$ buttons. The change-of-base formula lets you compute ANY logarithm using whichever base you have available. It works because all logarithms are proportional to each other—changing base just changes the scale factor.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: $\log_b x=\frac{\log_a x}{\log_a b}$ rewrites a log in whatever base you can actually compute.

Common stuck point: The procedure for change of base formula is the easy part; the trap is flipping the ratio. Asking "Is the base something other than $e$ or 10 that I must turn into a computable ratio of logs?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the base something other than $e$ or 10 that I must turn into a computable ratio of logs?

Worked Examples

Example 1

easy

Evaluate

\log_5(20)

using the change of base formula and a calculator.

Answer

\log_5(20) \approx 1.861

First step

The change of base formula:

\log_b(a) = \frac{\ln(a)}{\ln(b)}

(or equivalently

\frac{\log(a)}{\log(b)}

Full solution

2
Apply: $\log_5(20) = \frac{\ln(20)}{\ln(5)}$ .
3
Calculate: $\frac{\ln(20)}{\ln(5)} = \frac{2.9957}{1.6094} \approx 1.861$ .

The change of base formula converts a logarithm from any base to a quotient of logarithms in a base your calculator supports (usually base 10 or base

e

). This is essential because most calculators only have

\log

and

\ln

buttons.

Example 2

medium

Show that

\log_4(8) = \frac{3}{2}

using the change of base formula.

Example 3

medium

Use change-of-base to evaluate

\log_{25} 5

Example 4

medium

Solve

\log_2(x) = \log_4(x + 12)

for

x > 0

Example 5

medium

Show that

\log_b a = \dfrac{1}{\log_a b}

using change-of-base.

Example 6

hard

Simplify

\log_2 3 \cdot \log_3 4 \cdot \log_4 5 \cdot \log_5 8

Example 7

hard

Solve

\log_x 8 = 3

for

x

Example 8

hard

Show that

\log_{a^k} b = \dfrac{1}{k}\log_a b

for

k \ne 0

Example 9

challenge

Solve

\log_2 x \cdot \log_x 8 = \log_4 64

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Simplify

\log_9(27)

to an exact fraction.

Example 2

hard

Prove that

\log_a(b) \cdot \log_b(c) = \log_a(c)

using the change of base formula.

Example 3

easy

Rewrite

\log_2 7

using natural logarithms.

Example 4

easy

Rewrite

\log_5 20

using base-10 logarithms.

Example 5

easy

Evaluate

\log_3 9

without a calculator.

Example 6

easy

Evaluate

\log_b b

for any base

b>0

b\ne 1

Example 7

easy

Use change-of-base to write

\log_4 x

in terms of base 2 logs.

Example 8

easy

Rewrite

\log_7 50

as a ratio of natural logs.

Example 9

easy

Evaluate

\log_2 32

by recognizing a power.

Example 10

easy

Write

\log_{10} 7

as a natural-log ratio.

Example 11

medium

Evaluate

\log_4 64

exactly.

Example 12

medium

Show that

\log_2 5 \cdot \log_5 2 = 1

Example 13

medium

Express

\log_9 x

in terms of

\log_3 x

Example 14

medium

Given

\ln 2 \approx 0.693

and

\ln 3 \approx 1.099

, estimate

\log_2 3

Example 15

medium

Simplify

\frac{\log_5 16}{\log_5 4}

Example 16

medium

Solve

\log_4 x = \log_2 8

for

x

Example 17

medium

Write

\log_8 x

in terms of

\log_2 x

Example 18

medium

Evaluate

\log_2 3 \cdot \log_3 8

exactly.

Example 19

challenge

Prove

\log_a b = \frac{1}{\log_b a}

using change-of-base.

Example 20

challenge

\log_2 x = a

, express

\log_8 x

and

\log_4 x

in terms of

a

Example 21

challenge

Solve

\log_2 x + \log_4 x = 3

for

x

Example 22

medium

Evaluate

\log_{27} 81

exactly.

Example 23

easy

Evaluate

\log_2 64

without a calculator.

Example 24

easy

Use change-of-base to estimate

\log_2 10

to two decimals.

Example 25

easy

Use change-of-base to express

\log_8 3

using base-2 logs.

Example 26

easy

Rewrite

\log_5 x

as a ratio of base-

e

logarithms.

Example 27

medium

Use change-of-base to evaluate

\log_{27} 9

Example 28

medium

Approximate

\log_7 50

to three decimal places.

Example 29

medium

Simplify

\log_{16} 8

to an exact fraction.

Example 30

medium

Evaluate

\log_3 81 \cdot \log_9 27

exactly.

Example 31

medium

Solve

\log_3 x = \log_9 (x + 6)

for

x > 0

Example 32

medium

Evaluate

\log_{125} 25

exactly.

Example 33

hard

\log_2 5 = a

, express

\log_5 32

in terms of

a

Example 34

hard

\log_3 2 = p

and

\log_3 5 = q

, find

\log_5 6

in terms of

p

and

q

Example 35

hard

Solve

\log_2 x + \log_4 x = 6

for

x > 0

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Related Concepts

Logarithm Natural Logarithm Solving Exponential Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

logarithmnatural logarithm