Variables in the exponent
An exponential equation has the variable in an exponent:
- 2ˣ = 32
- 3^(x + 1) = 81
- 5000(1.04)ᵗ = 8000
Method 1: Common base
If both sides can be written as powers of the same base, equate exponents.
If bˣ = bʸ (b > 0, b ≠ 1), then x = y.
Example: 2ˣ = 32
- 32 = 2⁵
- x = 5
Example: 9^(x − 1) = 27
- Rewrite with a common base of 3: (3²)^(x − 1) = 3³
- 3^(2x − 2) = 3³
- 2x − 2 = 3 → x = 2.5
Example: 4ˣ = 1/8
- (2²)ˣ = 2⁻³ → 2x = −3 → x = −1.5
Useful powers to know
| Base | Powers |
|---|---|
| 2 | 2, 4, 8, 16, 32, 64, 128, 256 |
| 3 | 3, 9, 27, 81, 243 |
| 5 | 5, 25, 125, 625 |
| 10 | 10, 100, 1,000 |
Method 2: Logarithms
When no common base exists, take a logarithm of both sides.
Example: 3ˣ = 20
- log(3ˣ) = log 20
- x log 3 = log 20
- x = log 20 ÷ log 3 ≈ 2.73
Method 3: Graphing
Graph y = 3ˣ and y = 20; the intersection’s x-coordinate is the solution. Graphing technology makes this quick.
Applications
Exponential growth: A population of 5,000 grows 4% per year. When will it reach 8,000?
- 5,000(1.04)ᵗ = 8,000
- 1.04ᵗ = 1.6
- t = log 1.6 ÷ log 1.04 ≈ 12 years
Half-life: A substance’s half-life is 6 hours. How long until 100 mg becomes 12.5 mg?
- 100(½)^(t/6) = 12.5 → (½)^(t/6) = 1/8 = (½)³
- t/6 = 3 → t = 18 hours
Worked examples
Example 1: 5^(2x) = 125 → 2x = 3 → x = 1.5.
Example 2: 2^(x + 3) = 8^x → 2^(x + 3) = 2^(3x) → x + 3 = 3x → x = 1.5.
Example 3: 10ˣ = 450 → x = log 450 ≈ 2.65.
Common mistakes
- Equating exponents with different bases.
- Forgetting brackets when rewriting: (3²)^(x−1) = 3^(2x−2), not 3^(2x−1).
- Dividing the bases instead of using logs.
Try it yourself
- 3ˣ = 81
- 2^(x − 1) = 16
- 25ˣ = 5
- 4^(x + 1) = 8
- 2ˣ = 10 (to 2 decimals)
Answers: 1) 4 2) 5 3) 0.5 4) 0.5 5) ≈ 3.32
Quick recap
- Rewrite with a common base and equate exponents.
- Otherwise, use logarithms or graphing.
- Exponential equations model exponential growth and half-life.