Exponential and logarithmic functions are inverses of each other. If f(x) = bˣ, then its inverse is f⁻¹(x) = log_b(x). Understanding this relationship is key to solving equations involving both types.
Exponential Functions
f(x) = a · bˣ where b > 0 and b ≠ 1. When b > 1: exponential growth (population, compound interest). When 0 < b < 1: exponential decay (radioactive decay, cooling). The y-intercept is always (0, a) and the horizontal asymptote is y = 0.
Logarithmic Functions
f(x) = log_b(x) is defined only for x > 0. The x-intercept is always (1, 0) and the vertical asymptote is x = 0. The natural log ln(x) = log_e(x) is the most common in calculus.
Their Inverse Relationship
Because they are inverses, their graphs are reflections of each other across the line y = x. If b^y = x, then log_b(x) = y. This relationship is used to solve exponential equations: take the log of both sides.