Logarithm Properties Examples in Math

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

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 three fundamental rules of logarithms: the product rule $\log_b(xy) = \log_b x + \log_b y$ , the quotient rule $\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y$ , and the power rule $\log_b(x^n) = n\log_b x$ .

Logarithms were invented to turn hard operations into easy ones. Multiplication becomes addition, division becomes subtraction, and exponentiation becomes multiplication. This is why slide rules worked—they added lengths (logarithms) to multiply numbers.

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: Product becomes a sum, quotient a difference, and a power slides out front as a multiplier.

Common stuck point: The procedure for logarithm properties is the easy part; the trap is turning $\log(x+y)$ into $\log x+\log y$ . Asking "Is the log's argument a product, quotient, or power I can split into add, subtract, or a front multiplier?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the log's argument a product, quotient, or power I can split into add, subtract, or a front multiplier?

Worked Examples

Example 1

easy

Expand

\log_2(8x^3)

using logarithm properties.

Answer

3 + 3\log_2 x

First step

Apply the product rule:

\log_2(8x^3) = \log_2 8 + \log_2 x^3

Full solution

2
Evaluate $\log_2 8 = 3$ and apply the power rule: $\log_2 x^3 = 3\log_2 x$ .
3
Result: $3 + 3\log_2 x$ .

The product rule

\log_b(MN) = \log_b M + \log_b N

and power rule

\log_b(M^p) = p\log_b M

are the two most commonly used expansion properties.

Example 2

medium

Condense

2\ln x - \frac{1}{2}\ln(x + 1) + 3\ln y

into a single logarithm.

Example 3

medium

Use the change of base formula to evaluate

\log_3 20

to three decimal places.

Example 4

medium

Use the change-of-base formula to evaluate

\log_5 100

to three decimals.

Example 5

hard

Using

\log_b 2 = 0.43

\log_b 3 = 0.68

, and

\log_b 7 = 1.21

, find

\log_b 42

Practice Problems

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

Example 1

easy

Expand

\log\left(\frac{a^2}{b^5}\right)

Example 2

easy

Use the product rule to write

\log_2(8 \cdot 4)

as a sum.

Example 3

easy

Use the quotient rule to write

\log_3\!\left(\frac{27}{9}\right)

as a difference.

Example 4

easy

Use the power rule to rewrite

\log_5(x^4)

Example 5

easy

Evaluate

\log_7 1

Example 6

easy

Evaluate

\log_6 6

Example 7

easy

Write

\log_b m + \log_b n

as a single logarithm.

Example 8

easy

Write

\log_b p - \log_b q

as a single logarithm.

Example 9

easy

Rewrite

2\log_b x

using the power rule.

Example 10

medium

Expand

\log_b\!\left(\frac{x^2 y}{z}\right)

fully.

Example 11

medium

Condense

3\log_b x + \log_b y - 2\log_b z

into one logarithm.

Example 12

medium

Given

\log_b 2=0.43

and

\log_b 3=0.68

, find

\log_b 6

Example 13

medium

Given

\log_b 2=0.43

and

\log_b 3=0.68

, find

\log_b 8

Example 14

medium

Given

\log_b 2=0.43

and

\log_b 3=0.68

, find

\log_b 1.5

Example 15

medium

Solve

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

for

x

Example 16

medium

Simplify

\log_b\!\left(b^3\right) + \log_b 1

Example 17

medium

Express

\log_b\sqrt{xy}

in expanded form.

Example 18

medium

Given

\log_b 5=1.16

, find

\log_b 25

Example 19

challenge

Solve

\log_3(x+6) - \log_3 x = 2

for

x

Example 20

challenge

Prove that

\log_b(x^n)=n\log_b x

follows from the product rule for integer

n\ge 1

Example 21

challenge

Given

\log_b 2=0.43

and

\log_b 3=0.68

, find

\log_b 0.75

Example 22

easy

Use the product rule to expand

\log_b(5x)

Example 23

easy

Use the quotient rule to expand

\log_b\!\left(\dfrac{x}{7}\right)

Example 24

easy

Use the power rule to rewrite

\log_b(x^7)

Example 25

easy

Condense

\log_b 4 + \log_b 5

into a single logarithm.

Example 26

easy

Condense

\log_b 18 - \log_b 6

into a single logarithm.

Example 27

medium

Expand

\log_2\!\left(\dfrac{16 x^5}{y^3}\right)

fully.

Example 28

medium

Condense

\tfrac{1}{2}\ln x + 3\ln y - 2\ln z

into a single log.

Example 29

medium

Given

\log_b 2 = 0.43

and

\log_b 5 = 1.16

, find

\log_b 10

Example 30

medium

Given

\log_b 2 = 0.43

and

\log_b 5 = 1.16

, find

\log_b 2.5

Example 31

medium

Solve

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

for

x

Example 32

medium

Simplify

\log_3 81 + \log_3 9 - \log_3 27

Example 33

medium

Expand

\log\!\left(\dfrac{x^3 \sqrt{y}}{z^4}\right)

Example 34

medium

Given

\log_b 3 = 0.68

, find

\log_b 81

Example 35

medium

Solve

\log_5(x + 4) = \log_5(2x - 1)

Example 36

hard

Solve

\log_2(x - 1) + \log_2(x + 2) = 2

Example 37

hard

Solve

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

(base 10).

Example 38

hard

Solve

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

using logarithms.

Example 39

hard

Express

\log_2 3

in terms of natural logarithms.

Example 40

hard

Solve

\log_2(x) - \log_2(x - 2) = 3

Example 41

challenge

Prove

\log_b a \cdot \log_a b = 1

for any valid

a, b > 0,\ \ne 1

← Back to Logarithm Properties Formula Explained Practice Mode

Related Concepts

Logarithm Change of Base Formula Solving Exponential Equations Solving Logarithmic Equations

Background Knowledge

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

logarithm