An exponential function has the variable in the exponent, such as y = a · bˣ. If the base b is greater than 1, values grow faster and faster. If 0 < b < 1, values decay toward zero. That pattern models populations, investments, cooling, and some epidemics better than a straight line.
Reading the parameters
- a — starting amount when x = 0 (the vertical intercept for many models)
- b — growth factor per unit of x. For example b = 1.05 means “multiply by 1.05 each step” (about 5% growth)
- x — time or step count in the story
Linear change adds the same amount each time. Exponential change multiplies by the same factor. Mixing those ideas up is one of the most common precalculus mistakes.
What is a logarithm?
A logarithm answers: “To what power must I raise this base to get this number?” So log₂(8) = 3 because 2³ = 8. Logs are the inverse of exponential functions. If a model says “amount = 100 · 2ᵗ,” a log helps you solve for the time t when the amount hits a target.
Common bases
- Base 10 — common log, often written log
- Base e (≈ 2.718) — natural log, written ln; appears in continuous growth models
- Base 2 — natural for doubling processes
Graph sense
Exponential growth curves get steep. Exponential decay levels off toward a horizontal asymptote (often y = 0). Log graphs rise quickly at first for small positive x, then more slowly—and they are only defined for positive inputs.
Practice: Use Standard → Exponential & Log in the lab for story problems (savings, decay, doubling) with graphs. Start free practice.