Linear change adds the same amount each step; exponential change multiplies by the same factor. This unit builds from sequences to exponential functions and models, then turns them around: composition, inverses, and logarithms, the inverse of exponentials. It ends with semi-log plots, which straighten exponential data into a line.
Arithmetic sequences add a common difference; geometric sequences multiply by a common ratio. Linear functions are the continuous version of arithmetic sequences and exponential functions the continuous version of geometric ones. Equal differences over equal intervals mean linear; equal ratios mean exponential.
f(x) = abˣ with a ≠ 0, b > 0, b ≠ 1. With a > 0 it shows growth when b > 1 and decay when 0 < b < 1. It is always increasing or always decreasing, with one concavity and no extrema or inflection points. Exponent rules turn horizontal shifts into vertical dilations and horizontal dilations into a change of base.
Build an exponential model from an initial value and a ratio, or from two points. The base is the growth factor, tied to a percent change. Compare linear, quadratic and exponential models using context, and check a model with residuals: a residual plot with no pattern supports the model.
In f(g(x)), the outputs of g become the inputs of f. Composition is usually not commutative, the identity function x leaves a function unchanged, and complicated functions can be split into simpler pieces.
A function has an inverse on a domain where each output comes from exactly one input. The inverse swaps input-output pairs, swaps domain and range, reflects the graph over y = x, and undoes the function: f(f⁻¹(x)) = x.
log_b c is the exponent you put on b to get c. Logarithmic functions are inverses of exponential functions, so their graphs reflect over y = x. A log function has domain x > 0, range all reals, a vertical asymptote at x = 0, and no extrema or inflection points.
The product, power and change-of-base properties rewrite logs and explain their graphs. Use them, plus the inverse relationship with exponentials, to solve equations. Check for extraneous solutions, since log arguments must be positive.
Logarithmic models fit situations where inputs change proportionally over equal output intervals, like sound levels. On a semi-log plot (log-scaled y-axis), exponential data look linear, and the line's slope and intercept give the base and initial value.