Practice Logarithm Properties in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The three fundamental rules of logarithms: the product rule log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y, the quotient rule log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y, and the power rule log⁡b(xn)=nlog⁡bx\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.

Showing a random 20 of 50 problems.

Example 1

easy
Use the product rule to expand log⁡b(5x)\log_b(5x).

Example 2

medium
Given log⁡b5=1.16\log_b 5=1.16, find log⁡b25\log_b 25.

Example 3

medium
Expand log⁡ ⁣(x3yz4)\log\!\left(\dfrac{x^3 \sqrt{y}}{z^4}\right).

Example 4

challenge
Prove that log⁡b(xn)=nlog⁡bx\log_b(x^n)=n\log_b x follows from the product rule for integer n≥1n\ge 1.

Example 5

easy
Evaluate log⁡449\log_4 4^9.

Example 6

easy
Use the quotient rule to expand log⁡b ⁣(x7)\log_b\!\left(\dfrac{x}{7}\right).

Example 7

easy
Condense log⁡b18−log⁡b6\log_b 18 - \log_b 6 into a single logarithm.

Example 8

hard
Express log⁡23\log_2 3 in terms of natural logarithms.

Example 9

easy
Use the quotient rule to write log⁡3 ⁣(279)\log_3\!\left(\frac{27}{9}\right) as a difference.

Example 10

medium
Given log⁡b2=0.43\log_b 2=0.43 and log⁡b3=0.68\log_b 3=0.68, find log⁡b8\log_b 8.

Example 11

medium
Condense 12ln⁡x+3ln⁡y−2ln⁡z\tfrac{1}{2}\ln x + 3\ln y - 2\ln z into a single log.

Example 12

medium
Expand log⁡b ⁣(x2yz)\log_b\!\left(\frac{x^2 y}{z}\right) fully.

Example 13

medium
Given log⁡b2=0.43\log_b 2=0.43 and log⁡b3=0.68\log_b 3=0.68, find log⁡b6\log_b 6.

Example 14

easy
Condense log⁡b4+log⁡b5\log_b 4 + \log_b 5 into a single logarithm.

Example 15

medium
Solve log⁡2x+log⁡2(x+6)=4\log_2 x + \log_2(x + 6) = 4 for xx.

Example 16

medium
Given log⁡b2=0.43\log_b 2=0.43 and log⁡b3=0.68\log_b 3=0.68, find log⁡b1.5\log_b 1.5.

Example 17

easy
Use the power rule to rewrite log⁡b(x7)\log_b(x^7).

Example 18

medium
Rewrite log⁡ba=c\log_b a = c in exponential form.

Example 19

medium
Express log⁡bxy\log_b\sqrt{xy} in expanded form.

Example 20

easy
Use the power rule to rewrite log⁡5(x4)\log_5(x^4).
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