Practice Logarithm in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

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

The exponent that produces a number. log⁡2(8)=3\log_2(8) = 3 because 23=82^3 = 8.

Showing a random 20 of 50 problems.

Example 1

medium
Use log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y to expand log⁡2(8⋅16)\log_2(8 \cdot 16).

Example 2

medium
Evaluate log⁡100.001\log_{10} 0.001.

Example 3

easy
Evaluate log⁡327\log_3 27.

Example 4

medium
Simplify log⁡5100−log⁡54\log_5 100-\log_5 4.

Example 5

challenge
Solve log⁡2x+log⁡4x=3\log_2 x+\log_4 x=3.

Example 6

medium
Solve 2x=502^x = 50 for xx to two decimal places.

Example 7

easy
Evaluate ln⁡e\ln e.

Example 8

hard
Solve log⁡2(x)+log⁡2(x−6)=4\log_2(x) + \log_2(x - 6) = 4.

Example 9

easy
What is the domain of log⁡2x\log_2 x?

Example 10

easy
Evaluate log⁡28\log_2 8.

Example 11

easy
Rewrite 53=1255^3 = 125 as a logarithm.

Example 12

easy
Evaluate log⁡416\log_4 16.

Example 13

hard
What is the domain of f(x)=log⁡2(x−3)f(x) = \log_2(x - 3)?

Example 14

easy
Rewrite 24=162^4=16 as a logarithm.

Example 15

easy
What is log⁡b1\log_b 1 for any base bb?

Example 16

hard
Solve log⁡(x+1)+log⁡(x−1)=log⁡8\log(x+1) + \log(x-1) = \log 8 where log⁡=log⁡10\log = \log_{10}.

Example 17

medium
Solve log⁡3x=4\log_3 x=4.

Example 18

easy
Evaluate log⁡51\log_5 1.

Example 19

easy
Evaluate log⁡218\log_2 \tfrac{1}{8}.

Example 20

challenge
If log⁡b2=0.3\log_b 2=0.3 and log⁡b3=0.5\log_b 3=0.5, find log⁡b12\log_b 12.
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