Exponential Function
Functions & Advanced AlgebraAn exponential function has the form f(x) = a * 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 is one where the variable is in the exponent: $f(x) = b^x$. Each time $x$ increases by $1$, the output is multiplied by the same number (the base $b$).
Example
$f(x) = 2^x$: $f(0) = 1$, $f(1) = 2$, $f(2) = 4$, $f(3) = 8$. Every step doubles the output. This rapid growth is called "exponential growth."
Key Insight
Exponential functions grow or shrink by a constant multiplier, not a constant addition. Doubling is much faster than adding. A population doubling each year reaches $1{,}000$ in about $10$ years starting from $1$.
Definition
An exponential function has the form $f(x) = a \cdot b^x$, where $a$ is the initial value, $b > 0$, $b \neq 1$, and $x$ is the exponent. If $b > 1$, the function grows; if $0 < b < 1$, it decays. The domain is all reals; the range is all positive reals (when $a > 0$).
Example
A bank account with $\$500$ at $4\%$ annual interest: $A(t) = 500 \cdot (1.04)^t$. After $10$ years: $A(10) = 500 \cdot (1.04)^{10} \approx \$740.12$. The $y$-intercept is always $a$ (at $x = 0$, $b^0 = 1$).
Key Insight
The natural base $e \approx 2.718$ is special: the function $e^x$ is its own derivative. Exponential functions model compound interest, population growth, radioactive decay, and cooling.
Definition
The exponential function $\exp: \mathbb{C} \to \mathbb{C}$ is the unique solution to $f'(x) = f(x)$ with $f(0) = 1$. It can be 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
Solving $dy/dt = ky$ gives $y(t) = y_0 \cdot e^{kt}$. This underpins models in physics (nuclear decay), biology (population growth), and finance (continuous compounding: $A = Pe^{rt}$).
Key Insight
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.