Exponential Function
An exponential function is a mathematical function of the form f(x) = aˣ, where the variable x appears as the exponent and the base a is a positive constant not equal to 1. These functions exhibit rapid growth (when a > 1) or decay (when 0 < a < 1) and are fundamental models for population growth, compound interest, radioactive decay, and many natural phenomena.
Key Formula
f(x) = a^x, where a > 0 and a ≠ 1
LaTeX: f(x) = a^x, \quad a > 0,\; a \neq 1
| Symbol | Meaning | Unit |
|---|---|---|
| f(x) | Output value of the function | dimensionless |
| a | Base (positive constant) | dimensionless |
| x | Exponent (independent variable) | dimensionless |
Worked Example
Problem
A bacteria culture starts with 500 cells and doubles every hour. How many cells are present after 4 hours?
Solution
Step 1: The model is f(t) = 500 × 2ᵗ. Step 2: At t = 4: f(4) = 500 × 2⁴ = 500 × 16 = 8000.
Answer
8000 bacterial cells after 4 hours.
Properties of Exponential Functions f(x) = aˣ
| Property | a > 1 (Growth) | 0 < a < 1 (Decay) |
|---|---|---|
| Behaviour | Increasing | Decreasing |
| y-intercept | (0, 1) | (0, 1) |
| Horizontal asymptote | y = 0 (as x→−∞) | y = 0 (as x→+∞) |
| Domain | All real numbers | All real numbers |
| Range | (0, +∞) | (0, +∞) |
Interactive Tools
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Related Terms
Exponent
An exponent (also called a power or index) indicates how many times a base number is multiplied by itself. Written as aⁿ, where a is the base and n is the exponent, it represents repeated multiplication in a compact form. Exponents are essential in scientific notation, polynomial expressions, and exponential growth models.
Logarithmic Function
A logarithmic function is the inverse of an exponential function, defined as f(x) = log_b(x) for a positive base b ≠ 1 and positive input x. Its graph is a slowly increasing (or decreasing) curve that passes through (1, 0) and grows without bound, yet extremely slowly. Logarithmic functions model phenomena where large ranges of values need to be compressed, such as sound intensity, earthquake magnitude, and chemical acidity.
Geometric Sequence
A geometric sequence (or geometric progression) is an ordered list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio r. The nth term is given by aₙ = a₁ × r^(n−1). Geometric sequences model exponential growth and decay, compound interest, population doubling, and signal attenuation.
From Latin "exponere" (to set out) combined with "function" from Latin "functio" (performance). The special base e was identified by Jacob Bernoulli around 1683 and formalised by Leonhard Euler in the 18th century.