College Algebra

Change of Base

Lesson

Calculators only have $\log$ (base 10) and $\ln$ (base $e$ ). The change-of-base formula rewrites a log in any base as a ratio of logs in a base you can compute.

The formula

\log_b(x) = \frac{\log_c(x)}{\log_c(b)}

The new base $c$ can be anything you like — usually 10 (with $\log$ ) or $e$ (with $\ln$ ).

Worked example 1 — clean integer answer

\log_4(64) = \frac{\log_2(64)}{\log_2(4)} = \frac{6}{2} = 3

We picked base 2 because both 64 and 4 are nice powers of 2.

Worked example 2 — non-integer answer

\log_4(8) = \frac{\log_2(8)}{\log_2(4)} = \frac{3}{2}

Worked example 3 — using natural log

\log_5(20) = \frac{\ln 20}{\ln 5} \approx \frac{2.996}{1.609} \approx 1.861

When to use it

Both numbers are powers of the same base → use that base, get an exact answer.
Numbers aren’t nice powers → use $\ln$ or $\log$ for a decimal.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\log_2(8)

Problem 2

\log_3(27)

Problem 3

\log_5(125)

Problem 4

\log_2(16)

Practice

Standard problems matching the lesson.

Problem 5

\log_2(32)

Problem 6

\log_3(81)

Problem 7

\log_4(64)

Problem 8

\log_5(625)

Problem 9

\log_{10}(1000)

Problem 10

\log_2(1)

Problem 11

\log_7(49)

Problem 12

\log_3(243)

Problem 13

\log_2(64)

Problem 14

\log_8(512)

Problem 15

\log_9(81)

Problem 16

\log_2(2)

Problem 17

Signal halves each km; how many km to 1/8?

Problem 18

\log_5(1)

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

\log_{16}(64)

Problem 20

\log_4(8)

Problem 21

\log_{27}(9)

Problem 22

\log_{25}(125)

Problem 23

\log_8(16)

Problem 24

\log_4(2)

Problem 25

\log_9(3)

Ask the tutor

Stuck on a concept? Want another example? Ask anything about this topic.

Type your own question below, or tap one of the suggestions. The tutor can re-explain the lesson, work through a specific problem with you, generate fresh practice tuned to where you are, or check your reasoning.

Quiz

Test yourself on this topic →

10 questions, no hints. About 5 minutes.