Navigation: Wiki Home > Skills > Exponential Functions and Properties
Population growth, radioactive decay, compound interest, viral spread—these phenomena share a common pattern: the rate of change is proportional to the current amount. The mathematical tool that captures this is the exponential function.
Unlike polynomial functions where $x$ is raised to a power (like $x^2$ or $x^3$), in an exponential function the variable is in the exponent: $f(x) = b^x$. This seemingly small change produces dramatically different behavior.
Consider: if you fold a piece of paper 50 times (doubling its thickness each time), the final thickness would be about 17 million miles—enough to reach the sun! That's exponential growth.
| Concept | Formula/Property |
|---|---|
| Exponential Function | $f(x) = b^x$ where $b > 0$, $b \neq 1$ |
| Domain | $\mathbb{R}$ (all real numbers) |
| Range | $(0, \infty)$ (always positive) |
| Key Point | Always passes through $(0, 1)$ since $b^0 = 1$ |
| Increasing/Decreasing | $b > 1$: increasing; $0 < b < 1$: decreasing |
$$\boxed{f(x) = b^x \text{ is an \textbf{exponential function} where } b > 0 \text{ and } b \neq 1}$$
The constant $b$ is called the base. The variable $x$ is the exponent.
Crucial distinction:
These behave very differently! Exponential functions eventually outgrow any polynomial.
We build up the meaning step by step:
| Type of Exponent | Meaning | Example |
|---|---|---|
| Positive integer $n$ | $b^n = b \cdot b \cdot \ldots \cdot b$ ($n$ factors) | $2^3 = 2 \cdot 2 \cdot 2 = 8$ |
| Zero | $b^0 = 1$ | $5^0 = 1$ |
| Negative integer $-n$ | $b^{-n} = \frac{1}{b^n}$ | $2^{-3} = \frac{1}{8}$ |
| Rational $\frac{p}{q}$ | $b^{p/q} = \sqrt[q]{b^p} = (\sqrt[q]{b})^p$ | $8^{2/3} = (\sqrt[3]{8})^2 = 4$ |
| Irrational | Limit of rational approximations | $2^{\sqrt{3}} \approx 3.322$ |
For irrational exponents like $2^{\sqrt{3}}$, we use the fact that $\sqrt{3} = 1.732050808...$:
$$2^{\sqrt{3}} = \lim_{r \to \sqrt{3}} 2^r \quad \text{where } r \text{ is rational}$$
The values $2^{1.7}, 2^{1.73}, 2^{1.732}, ...$ converge to a unique real number.
$$\boxed{\begin{aligned} &1. \quad b^{x+y} = b^x \cdot b^y \\[0.3em] &2. \quad b^{x-y} = \frac{b^x}{b^y} \\[0.3em] &3. \quad (b^x)^y = b^{xy} \\[0.3em] &4. \quad (ab)^x = a^x \cdot b^x \end{aligned}}$$
These laws, familiar from algebra with rational exponents, extend to all real exponents.
y y y
| / | |
| / |------ y = 1 | \
| / | | \
| / | | \
|_/________________________ |_______________________ |____\________________
| x | x | x
(0,1) (0,1) (0,1)
b > 1 (increasing) b = 1 (constant) 0 < b < 1 (decreasing)Key observations:
$$\boxed{\begin{aligned} &\text{If } b > 1: \quad \lim_{x \to \infty} b^x = \infty \quad \text{and} \quad \lim_{x \to -\infty} b^x = 0 \\[0.5em] &\text{If } 0 < b < 1: \quad \lim_{x \to \infty} b^x = 0 \quad \text{and} \quad \lim_{x \to -\infty} b^x = \infty \end{aligned}}$$
In either case, the $x$-axis is a horizontal asymptote.
| Base | Behavior | Example Uses |
|---|---|---|
| $b > 1$ | Exponential growth | Population, compound interest |
| $b = 1$ | Constant function $y = 1$ | Not interesting! |
| $0 < b < 1$ | Exponential decay | Radioactive decay, cooling |
Larger bases grow faster: $10^x$ grows faster than $2^x$ for $x > 0$.
| Mistake | Correct Understanding |
|---|---|
| Thinking $b^0 = 0$ | $b^0 = 1$ for any $b > 0$ |
| Confusing $b^{x+y}$ with $b^x + b^y$ | $b^{x+y} = b^x \cdot b^y$ (multiply, don't add) |
| Writing $(ab)^x = a^x b$ | $(ab)^x = a^x \cdot b^x$ (both get exponent) |
| Thinking $b^x$ can be negative | $b^x > 0$ always (with $b > 0$) |
| Confusing $2^x$ and $x^2$ | Exponential vs. power function—very different! |
Evaluate without a calculator: (a) $4^{3/2}$ (b) $27^{-2/3}$ (c) $9^0 + 9^1$
Simplify using the laws of exponents: $$\frac{3^{x+2} \cdot 3^{2x}}{3^{x-1}}$$
Starting with the graph of $y = 2^x$, describe the transformations needed to obtain the graph of $y = 2^{-x} - 1$. Then identify the horizontal asymptote.
A scientist measures a quantity at two times and records:
Find constants $A$ and $b$ so that $Q(t) = A \cdot b^t$ models this data.
Evaluate: $$\lim_{x \to \infty} \frac{5^x - 3^x}{5^x + 3^x}$$
Show that $2^x$ eventually exceeds $x^{10}$. That is, prove there exists some $N$ such that $2^x > x^{10}$ for all $x > N$.
Hint: Consider the ratio $\frac{2^x}{x^{10}}$ and what happens as $x \to \infty$.
Question 1: The function $f(x) = 3^x$ passes through which of the following points?
(A) $(0, 0)$ (B) $(0, 1)$ (C) $(1, 1)$ (D) $(3, 1)$
(B) For any exponential function $b^x$, we have $b^0 = 1$, so the graph passes through $(0, 1)$. It does NOT pass through the origin.
Question 2: Which grows faster as $x \to \infty$: $f(x) = 1000x^{100}$ or $g(x) = 1.001^x$?
(A) $f(x) = 1000x^{100}$ grows faster (B) $g(x) = 1.001^x$ grows faster (C) They grow at the same rate (D) It depends on the value of $x$
(B) Even though $1.001$ is barely larger than 1, the exponential $1.001^x$ eventually dominates any polynomial. Exponential growth always beats polynomial growth for large $x$, regardless of coefficients or degrees.
Question 3: If $f(x) = b^x$ where $0 < b < 1$, which statement is TRUE?
(A) $f$ is increasing and $f(x) > 0$ for all $x$ (B) $f$ is decreasing and $f(x) > 0$ for all $x$ (C) $f$ is decreasing and $f(x) < 0$ for some $x$ (D) $f$ has a horizontal asymptote at $y = 1$
(B) When $0 < b < 1$, the function is decreasing (each multiplication by a fraction less than 1 makes the value smaller). However, $b^x > 0$ always—exponential functions never produce negative values.
The Multiplier Effect:
Think of $b^x$ as starting with 1 and multiplying by $b$ a total of $x$ times.
This is why $b^x > 0$ always: you start positive and multiply by positive numbers.
Real-world analogy: Compound interest at rate $r$ multiplies your money by $(1 + r)$ each year. After $t$ years: $A = A_0(1+r)^t$. The "multiplier effect" makes even small rates produce dramatic long-term growth.
| Previous | Up | Next |
|---|---|---|
| One-to-One Functions | Skills Index | The Number e |
Last updated: 2026-01-23