Logarithm Examples: 46 Problems with Answers

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

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

Concept Recap

The logarithm $\log_b(x)$ answers: "to what power must $b$ be raised to produce $x$ ?" It is the inverse function of $f(x) = b^x$ .

The exponent that produces a number. $\log_2(8) = 3$ because $2^3 = 8$ .

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: A logarithm answers what power the base must be raised to in order to reach a given number.

Common stuck point: The procedure for logarithm is the easy part; the trap is treating a log as division. Asking "Am I asking 'what exponent on the base gives this number?'" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asking 'what exponent on the base gives this number?'

Worked Examples

Example 1

easy

Evaluate

\log_2 32

Answer

5

First step

A logarithm asks for the exponent, so we want the value of

x

such that

2^x = 32

Full solution

2
Check powers of 2: $2^5 = 32$ .
3
Therefore $\log_2 32 = 5$ .

A logarithm answers the question: 'What power do I raise the base to in order to get this number?' The definition

\log_b a = c

means

b^c = a

Example 2

medium

Solve

\log_3(2x + 1) = 4

Example 3

hard

Solve

\log_2(x) + \log_2(x - 6) = 4

Example 4

medium

Solve

2^x = 50

for

x

to two decimal places.

Example 5

medium

Solve

\log_5 x = 3

Example 6

hard

Solve

3^{2x+1} = 27

Example 7

hard

A population doubles every 7 years. If it starts at

5000

, how many years until it reaches

40000

Example 8

hard

Solve

\log_3(x) + \log_3(x+6) = 3

Example 9

challenge

Carbon-14 has a half-life of 5730 years. A sample retains 30% of its original

^{14}

C. How old is it (to the nearest 100 years)?

Practice Problems

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

Example 1

easy

Evaluate

\log_5 125

Example 2

medium

Solve

\log(x) + \log(x - 3) = 1

where

\log

denotes

\log_{10}

Example 3

easy

Evaluate

\log_2 8

Example 4

easy

Evaluate

\log_{10} 1000

Example 5

easy

Evaluate

\log_5 1

Example 6

easy

Evaluate

\log_3 27

Example 7

easy

Rewrite

2^4=16

as a logarithm.

Example 8

easy

Evaluate

\ln e

Example 9

easy

Evaluate

\log_2 \frac{1}{4}

Example 10

easy

What is the domain of

\log_2 x

Example 11

medium

Simplify

\log_2 4+\log_2 8

Example 12

medium

Solve

\log_3 x=4

Example 13

medium

Simplify

\log_5 100-\log_5 4

Example 14

medium

Simplify

\log_2 8^5

Example 15

medium

Solve

\log_2(x-1)=3

Example 16

medium

Use the change of base to write

\log_4 9

with natural logs.

Example 17

medium

Solve

\log x+\log(x-3)=1

(base 10).

Example 18

medium

Why is

\log(2+3)

not equal to

\log 2+\log 3

Example 19

challenge

Solve

\log_2 x+\log_4 x=3

Example 20

challenge

\log_b 2=0.3

and

\log_b 3=0.5

, find

\log_b 12

Example 21

challenge

Solve

2^{2x}=3\cdot 2^x+4

using logs/substitution.

Example 22

medium

Solve

\log_2 x=-3

Example 23

easy

Evaluate

\log_4 16

Example 24

easy

Evaluate

\log_{10} 100000

Example 25

easy

Rewrite

5^3 = 125

as a logarithm.

Example 26

easy

Evaluate

\log_2 \tfrac{1}{8}

Example 27

medium

Use

\log_b(xy) = \log_b x + \log_b y

to expand

\log_2(8 \cdot 16)

Example 28

medium

Use the quotient rule to evaluate

\log_3 \tfrac{81}{9}

Example 29

medium

Use the power rule: evaluate

\log_2(8^5)

Example 30

medium

Solve

\log_2(x - 1) = 5

Example 31

medium

Write

3\log x - \log y

as a single logarithm.

Example 32

medium

Evaluate

\log_{10} 0.001

Example 33

hard

Solve

\log(x+1) + \log(x-1) = \log 8

where

\log = \log_{10}

Example 34

hard

Solve

5^{x} = 2 \cdot 5^{x-1} + 75

Example 35

hard

Solve

\ln(x) = 2

Example 36

hard

\log_2 3 = a

, express

\log_2 12

in terms of

a

Example 37

challenge

Solve

(\log_2 x)^2 - 3\log_2 x + 2 = 0

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Related Concepts

Exponential Function Inverse Function Natural Logarithm

Background Knowledge

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

exponential functioninverse function