Algebra II

Solving Exponential and Logarithmic Equations

Lesson

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

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

2^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.

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

Worked example 1 — exponential

3^x = 81

Rewrite 81 as a power of 3:

3^x = 3^4 \;\Rightarrow\; x = 4

Worked example 2 — logarithmic

\log_2(x) = 5

Rewrite as exponential:

x = 2^5 = 32

Worked example 3 — log of expression

\log_3(x + 5) = 2

x + 5 = 3^2 = 9

x = 4

Work through these. Stuck? Click Get a hint.

Quick problems to get going.

Problem 1

2^x = 8

Problem 2

3^x = 9

Problem 3

\log_2(x) = 3

Problem 4

\log_3(x) = 2

Standard problems matching the lesson.

Problem 5

2^x = 32

Problem 6

3^x = 81

Problem 7

5^x = 125

Problem 8

10^x = 10000

Problem 9

\log_2(x) = 4

Problem 10

\log_5(x) = 3

Problem 11

\log_4(x) = 2

Problem 12

2^{x + 1} = 16

Problem 13

3^{x - 2} = 27

Problem 14

\log_3(x + 5) = 2

Problem 15

\log_2(x - 1) = 5

Problem 16

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

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

Problem 17

5^{2x} = 125

Problem 18

4^{x + 3} = 64

Problem 19

2^{x + 1} = 32

Problem 20

\log_2(x) + \log_2(4) = 5

Problem 21

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

Problem 22

9^x = 27