Practice Solving Logarithmic Equations in Math

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

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Quick Recap

Solving equations containing logarithms by converting to exponential form or using log properties to combine and simplify.

If logarithms trap the variable inside a log⁡\log, converting to exponential form releases it. The key insight is that log⁡b(stuff)=c\log_b(\text{stuff}) = c means bc=stuffb^c = \text{stuff}—just rewrite and solve.

Showing a random 20 of 50 problems.

Example 1

easy
Why must we reject x=−3x = -3 when solving an equation containing log⁡x\log x?

Example 2

challenge
Solve (log⁡2x)2−3log⁡2x+2=0(\log_2 x)^2 - 3\log_2 x + 2 = 0.

Example 3

hard
Solve log⁡2(log⁡3x)=1\log_2(\log_3 x) = 1.

Example 4

medium
Solve log⁡3(x+6)−log⁡3x=2\log_3(x+6) - \log_3 x = 2.

Example 5

easy
Solve log⁡2(x)=5\log_2(x) = 5.

Example 6

medium
Solve log⁡5(x2−24)=2\log_5(x^2 - 24) = 2.

Example 7

hard
Solve log⁡3(x+5)+log⁡3(x−3)=2\log_3(x+5) + \log_3(x-3) = 2.

Example 8

medium
Solve log⁡2(x+4)−log⁡2(x−1)=3\log_2(x+4) - \log_2(x-1) = 3.

Example 9

medium
Solve ln⁡(x2−4)=ln⁡(3x)\ln(x^2 - 4) = \ln(3x).

Example 10

easy
Solve log⁡2x=−1\log_2 x = -1.

Example 11

medium
Solve 2log⁡5x=log⁡5812\log_5 x = \log_5 81.

Example 12

easy
log⁡b(b)=\log_b(b) = ___.

Example 13

easy
Solve log⁡2x=4\log_2 x = 4.

Example 14

easy
Solve log⁡3(2x)=2\log_3(2x) = 2.

Example 15

challenge
Solve log⁡2x+log⁡4x+log⁡8x=11\log_2 x + \log_4 x + \log_8 x = 11.

Example 16

medium
Solve log⁡6(x)+log⁡6(x−1)=1\log_6(x) + \log_6(x-1) = 1.

Example 17

easy
Solve log⁡3(9)=x\log_3(9) = x.

Example 18

medium
Solve ln⁡(x)=2\ln(x) = 2, leaving the answer in terms of ee.

Example 19

easy
Solve log⁡2x=3\log_2 x = 3.

Example 20

medium
Express log⁡b ⁣(x2yz)\log_b\!\left(\dfrac{x^2 y}{z}\right) as a sum/difference of logs.
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