Exponential Function Properties

Exponential Function Properties

Exponential functions have many of the same properties as any other exponent. Here's a quick overview with how they relate to functions.

Growth Vs. Decay

Exponential functions in the form $f(x)=a\cdot b^x$ will either show exponential growth or exponential decay. If $b>1$, you'll see exponential growth.

If $0<b<1$, you'll see exponential decay.

Exponent Rules

Exponent rules are used to multiply, divide, and otherwise work with exponents.

Product Rule: If you multiply two exponents with the same base, you add together the exponents. $$2^x\cdot 2^{3x}=2^{4x}$$

Quotient Rule: Similar to the product rule, if you divide two exponents with the same base, you subtract the exponents. $$\frac{2^{4x}}{2^x}=2^{3x}$$

Power Rule: If you take an exponent to a power, you multiply the two powers. $$(3^{2x})^3=3^{6x}$$

Zero-Exponent Rule: A number taken to a power of $0$ is $1$ $$b^0=1$$

Negative Exponent Rule: A number taken to a negative exponent can instead be put in the denominator as a positive exponent. $$3^{-3x}=\frac{1}{3^{3x}}$$

We can use these exponent rules to manipulate exponential functions to make them easier to work with. Let's look at how we can break down the function $f(x)=(e^{2x+1}\cdot e^{-x-3})^2$

$$\begin{align}f(x)&=(e^{2x+1}\cdot e^{-x-3})^2\\ f(x)&=(e^{2x+1-x-3})^2\\ f(x)&=(e^{x-2})^2\\ f(x)&=e^{2(x-2)}\\ f(x)&=e^{2x-4}\\ f(x)&=\frac{e^{2x}}{e^4}\end{align}$$

Any of the steps in this problem will give you the same graph!

Rate of Change

The rate of change in an exponential function is constantly changing. If the function shows exponential growth, the rate of change is constantly increasing. If the function shows exponential decay, the rate of change is constantly decreasing.