What Is an Exponential Function?

There's a specific type of function where the variable sits in the exponent — not as the base. These are called exponential functions, and they behave in a dramatically different way from the polynomial functions you've probably seen before.

What Is an Exponential Function?

Consider these two expressions side by side:

$$x^2 \quad \text{vs.} \quad 2^x$$

In the first, $x$ is the base and 2 is the exponent. That's a polynomial — you've worked with these before.

In the second, 2 is the base and $x$ is the exponent. That's an exponential function. The variable is up in the power position.

This one swap — variable in the exponent instead of the base — changes everything about how the function behaves. Polynomial functions eventually level off relative to exponential functions. Exponential functions can grow (or shrink) at a breathtaking rate as x increases.

The general form of an exponential function is:

$$y = b^x$$

where $b$ is a positive constant called the base, and $b \neq 1$. (If $b = 1$, then $1^x = 1$ no matter what x is — a flat line, not very interesting.)

Growth vs. Decay

The value of the base determines whether the function grows or decays:

If $b > 1$: the function increases as x increases. This is exponential growth. The larger the base, the faster the growth.
If $0 < b < 1$: the function decreases as x increases. This is exponential decay. Think of a fraction raised to higher and higher powers — it keeps getting smaller.

For example:

$y = 2^x$ grows: $2^1 = 2$, $2^2 = 4$, $2^3 = 8$, $2^{10} = 1024$
$y = \left(\frac{1}{2}\right)^x$ decays: $\left(\frac{1}{2}\right)^1 = 0.5$, $\left(\frac{1}{2}\right)^2 = 0.25$, $\left(\frac{1}{2}\right)^{10} \approx 0.001$

One important property both share: no matter how far left you go on the x-axis (very negative x), the function never actually reaches zero — it just gets closer and closer. The x-axis is a horizontal asymptote.

Evaluating Exponential Functions

Evaluating an exponential function works the same as any other function: substitute the given x-value and compute.

Example 1

Evaluate $f(x) = 3^x$ for $x = 4$.

$$f(4) = 3^4 = 3 \times 3 \times 3 \times 3 = 81$$

Example 2

Evaluate $f(x) = 2^x$ for $x = -3$.

$$f(-3) = 2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125$$

Negative exponents flip to the denominator — a useful reminder that exponential functions with $b > 1$ are still positive for negative x, just small.

Example 3

Evaluate $f(x) = 5 \cdot 2^x$ for $x = 3$.

$$f(3) = 5 \cdot 2^3 = 5 \cdot 8 = 40$$

The coefficient in front of the exponential (the 5 here) scales the output but doesn't change the base or the growth rate.

Real-World Applications

Exponential functions show up constantly in science, finance, and everyday life. Whenever something grows or shrinks by a fixed percentage over equal time intervals, you're looking at exponential behavior.

Population and bacteria growth

If a bacteria colony starts with 100 cells and grows by 12% each hour, its population after $t$ hours is:

$$B(t) = 100 \cdot 1.12^t$$

The 1.12 comes from "100% of what you had plus 12% more." After 4 hours:

$$B(4) = 100 \cdot 1.12^4 = 100 \cdot 1.5735 \approx 157$$

After just 4 hours, the colony is 57% larger than it started.

Compound interest

Money in a savings account earning interest works the same way. If you deposit $1,000 at 5% annual interest, your balance after $t$ years is:

$$A(t) = 1000 \cdot 1.05^t$$

After 10 years: $A(10) = 1000 \cdot 1.05^{10} \approx $1{,}629$. After 30 years: $A(30) = 1000 \cdot 1.05^{30} \approx $4{,}322$. This is why compound interest is so powerful — and why waiting pays off.

Radioactive decay

Radioactive materials decay exponentially. If a substance has a half-life of 10 years, meaning half of it disappears every 10 years, and you start with 80 grams:

$$A(t) = 80 \cdot \left(\frac{1}{2}\right)^{t/10}$$

After 30 years: $A(30) = 80 \cdot \left(\frac{1}{2}\right)^3 = 80 \cdot \frac{1}{8} = 10$ grams remain.

Practice Problems

Evaluate $f(x) = 4^x$ for $x = 3$.

Show answer$f(3) = 4^3 = 64$

Evaluate $g(x) = 3 \cdot 2^x$ for $x = 5$.

Show answer$g(5) = 3 \cdot 2^5 = 3 \cdot 32 = 96$

Is $f(x) = 5^x$ an example of exponential growth or decay?

Show answerGrowth. The base is 5, which is greater than 1, so the function increases as x increases.

A town's population is modeled by $P(t) = 4000 \cdot 1.03^t$, where $t$ is years from now. What will the population be in 5 years?

Show answer$P(5) = 4000 \cdot 1.03^5 = 4000 \cdot 1.1593 \approx 4{,}637$ people.

A car worth $25,000 depreciates by 15% per year. Write an exponential function for its value after $t$ years, then find its value after 4 years.

Show answerLosing 15% per year means keeping 85%, so the base is 0.85: $V(t) = 25000 \cdot 0.85^t$. After 4 years: $V(4) = 25000 \cdot 0.85^4 = 25000 \cdot 0.5220 \approx $13{,}050$.

You can graph exponential functions using our Equation Grapher to see the growth or decay curve for yourself.

What Is an Exponential Function?