Exponential and Logarithmic Functions
Fifteen topics on multiplicative change and how to undo it. Arithmetic and geometric sequences, exponential functions built on a constant factor, composition and inverse functions, the logarithm as the exponent question, log properties and equation solving with domain discipline, data modeling with exponentials and logs, and reading semi-log plots.
Topics
Key forms For every problem in this unit
Arithmetic sequence
an = a0 + d·n (constant difference d)
Geometric sequence
gn = g0 · rn (constant ratio r)
Exponential function
f(x) = a · bx, a = initial value, b = growth factor
Factor vs rate
b = 1 + r (5% growth → b = 1.05; 5% decay → b = 0.95)
Log means exponent
logb(c) = a ⇔ ba = c
Product rule
logb(xy) = logbx + logby
Power rule
logb(xk) = k·logbx
Inverse pair
f(f−1(x)) = x on the appropriate domain; graphs reflect over y = x
No rule exists for
logb(x + y) — it does not split