Exponential Function
An exponential function has the form $f(x) = a \cdot b^x$, where the variable appears as the exponent and the base $b$ is a positive constant not equal to $1$.
Formula
f(x) = a \cdot b^x
Definition
An exponential function has the variable in the exponent: $f(x) = a \cdot b^x$, where $a$ is the initial value, $b > 0$, $b \neq 1$, and $x$ is the exponent; each time $x$ increases by $1$, the output is multiplied by the base $b$, growing if $b > 1$ and decaying if $0 < b < 1$. The domain is all reals and the range is all positive reals (for $a > 0$). Formally, the exponential function $\exp: \mathbb{C} \to \mathbb{C}$ is the unique solution to $f'(x) = f(x)$ with $f(0) = 1$, defined by the power series $\exp(x) = \sum_{n=0}^{\infty} x^n / n!$; Euler's identity $e^{i\pi} + 1 = 0$ connects it to trigonometry via $\exp(ix) = \cos(x) + i\sin(x)$.
Example
$f(x) = 2^x$ gives $f(0) = 1$, $f(1) = 2$, $f(2) = 4$, $f(3) = 8$, doubling every step, which is called exponential growth. A bank account with $\$500$ at $4\%$ annual interest, $A(t) = 500 \cdot (1.04)^t$, grows to about $\$740.12$ after $10$ years, with the $y$-intercept always equal to $a$ (since $b^0 = 1$). Solving $dy/dt = ky$ gives $y(t) = y_0 \cdot e^{kt}$, underpinning models in physics (nuclear decay), biology (population growth), and finance (continuous compounding, $A = Pe^{rt}$).
Key Insight
Exponential functions grow or shrink by a constant multiplier rather than a constant addition, so doubling is much faster than adding: a population doubling each year reaches $1{,}000$ in about $10$ years starting from $1$. The natural base $e \approx 2.718$ is special because $e^x$ is its own derivative, which is why exponential functions model compound interest, population growth, radioactive decay, and cooling. The exponential map generalizes from $\mathbb{R}$ to Lie groups, $\exp: g \to G$ maps the Lie algebra to the group, encoding infinitesimal symmetries as global transformations.