AP Precalculus · Lesson 2 of 4
Exponential and logarithmic functions
Read an initial amount and a growth factor, and use logarithms to solve for an exponent.
Exponential models
In P(t) = 200(1.15)^t, the initial amount is 200 because any nonzero number to the power 0 is 1. Each increase of 1 in t multiplies the amount by 1.15, which is a 15% increase, not a 115% increase. The 1 in 1.15 is the original amount kept, and the 0.15 is the increase.
Equal time steps with a constant ratio support an exponential model. Equal time steps with a constant difference support a linear model. Check the ratio or the difference before choosing a formula.
Logarithms
A logarithm asks for an exponent. If 2^x = 16, then x = log₂(16) = 4 because 2⁴ = 16. The input of a logarithm must be positive, so the domain is part of the answer when the input is an expression.
To solve 3(2)^t = 48, divide by 3 first: 2^t = 16, so t = 4. Taking a logarithm before isolating the exponential term usually creates extra algebra.
Worked example
A population is P(t) = 500(0.8)^t. What does 0.8 mean, and what percent change is that per time unit?
- At t = 0, P = 500.
- Each time unit multiplies the population by 0.8.
- It keeps 80% of the previous amount, which is a 20% decrease per time unit.
Why this works. A factor between 0 and 1 is decay. Subtract the factor from 1 to get the percent decrease, as a decimal, when the factor is positive.
Check your understanding
- Read the initial amount by substituting t = 0.
- Convert a growth factor to a percent increase or decrease.
- Isolate the exponential before using a logarithm, and keep the logarithm’s domain.