When the unknown is in the power, take logs of both sides. 1.5ˣ = 174 becomes x log 1.5 = log 174, so x = log 174 ÷ log 1.5 = 12.7 to 3 s.f.
If both sides can be written with the same base, compare the powers instead: 4ˣ⁺⁴ = 2ˣ becomes 2²ˣ⁺⁸ = 2ˣ, so 2x + 8 = x and x = −8. Exponential models such as P = 850eᵏᵗ are solved the same way, using ln.
Worked example
Solve 3ˣ = 20 to 3 s.f.
- Take logs: x log 3 = log 20.
- x = log 20 ÷ log 3 = 2.726…
Answer:x = 2.73
Key words
- exponential equation
- an equation with the unknown in the power
- ln
- the natural logarithm, base e
- model
- an equation describing a real situation
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Common Core-referenced: HSF-LE.A.4