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Algebra II

Solving Exponential and Logarithmic Equations

Lesson

Logs and exponents are inverses, so we use one to undo the other.

Exponential equations

For an equation like 2x=162^x = 16: rewrite the right side with the same base, then match exponents.

2x=16  ⇒  2x=24  ⇒  x=42^x = 16 \;\Rightarrow\; 2^x = 2^4 \;\Rightarrow\; x = 4

When the bases can’t be matched as nice powers, take a log of both sides — but in this practice topic we’ll stick to clean cases.

Logarithmic equations

For log⁡b(x)=y\log_b(x) = y, rewrite as the exponential x=byx = b^y and solve.

Worked example 1 — exponential

3x=813^x = 81

Rewrite 81 as a power of 3:

3x=34  ⇒  x=43^x = 3^4 \;\Rightarrow\; x = 4

Worked example 2 — logarithmic

log⁡2(x)=5\log_2(x) = 5

Rewrite as exponential:

x=25=32x = 2^5 = 32

Worked example 3 — log of expression

log⁡3(x+5)=2\log_3(x + 5) = 2
x+5=32=9x + 5 = 3^2 = 9
x=4x = 4

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

2x=82^x = 8

Problem 2

3x=93^x = 9

Problem 3

log⁡2(x)=3\log_2(x) = 3

Problem 4

log⁡3(x)=2\log_3(x) = 2

Practice

Standard problems matching the lesson.

Problem 5

2x=322^x = 32

Problem 6

3x=813^x = 81

Problem 7

5x=1255^x = 125

Problem 8

10x=1000010^x = 10000

Problem 9

log⁡2(x)=4\log_2(x) = 4

Problem 10

log⁡5(x)=3\log_5(x) = 3

Problem 11

log⁡4(x)=2\log_4(x) = 2

Problem 12

2x+1=162^{x + 1} = 16

Problem 13

3x−2=273^{x - 2} = 27

Problem 14

log⁡3(x+5)=2\log_3(x + 5) = 2

Problem 15

log⁡2(x−1)=5\log_2(x - 1) = 5

Problem 16

2x=182^x = \tfrac{1}{8}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

52x=1255^{2x} = 125

Problem 18

4x+3=644^{x + 3} = 64

Problem 19

2x+1=322^{x + 1} = 32

Problem 20

log⁡2(x)+log⁡2(4)=5\log_2(x) + \log_2(4) = 5

Problem 21

log⁡3(x)−log⁡3(2)=2\log_3(x) - \log_3(2) = 2

Problem 22

9x=279^x = 27

Practice

Standard problems matching the lesson.

Problem 23

Bacteria double every hour; after x hours, 32× the start. Solve 2^x = 32.

Problem 24

Investment triples each year. Solve 3^x = 81 for x.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Sound: 10·log(x) = 30 (log base 10). Find x.

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