What Is an Exponential Function?

Summary:

• What is an exponent? An exponent is the amount of times a number is multiplied by itself. In the equation b³, 3 is the exponent, signifying b x b x b.
• What is an exponential function? Exponential functions measure exponential growth or decay via repeated multiplication.
• How to write exponential functions: You can write an exponential function in many ways. A simple format is f(x) = b^x.
• What are some examples of exponential function? Examples of exponential functions include f(x) = 3^x and f(x) = 0.25^x.

What Is an Exponent?

What is the definition of an exponent? In mathematics, an exponent signifies the amount of times a number is multiplied by itself. 

How do you convert a multiplication expression into exponential form? Exponents, also known as power, are written as a superscript to the right of a base number, e.g. b³, or b (the base number) to the power of 3 (the exponent). Here, the exponent expresses repeated multiplications: b multiplied by itself three times.

What is an Exponential Function?

An exponential function¹ is used to calculate exponential growth or decay via a simple formula. As for how to write exponential functions, the form is f(x) = b^x, where a is a fixed positive real number not equal to 1, and x, the exponent, is a real variable.

Core Elements of Exponential Functions

Let’s return to our example of an exponential function: f(x) = b^x. The elements of this exponential function are: 

• The domain of f is all real numbers (–∞, ∞)
• The range of f is all positive real numbers (0, ∞)
• b, the base of the exponential function,is a real number—greater than zero, and not equaling 1
• x represents any real number

Exponential functions have a constant base, as the variable is in the exponent. In other words, the rate of growth or decay is proportional to how much growth or decay is already there.²

There’s also the natural exponential function, which is when the base is the irrational number e (e ≈ 2.71828). The natural exponential function is written as exp(x). 

Exponential Functions Examples and Equations

There are increasing exponential functions and decreasing exponential functions.

If b > 1, f(x) = b^x is an increasing function. Increasing functions measure exponential growth. Examples of exponential functions that measure growth might look like:

f(x) = 3^x

If 0 < b < 1, f(x) = b^x is an decreasing function. Decreasing functions, naturally, measure exponential decay. Examples of exponential functions that measure growth might look like: Examples of exponential functions that measure growth might look like:

f(x) = 0.25^x

How to Graph Exponential Functions

In increasing exponential functions, the function grows very quickly to the right and decays very quickly to the left.²

In decreasing exponential functions, the function decays quickly to the right and grows quickly to the left. 

Exponential Functions: Practical Applications

Exponential functions have a variety of real-world, practical applications, from calculating compound interest in finance to modeling population dynamics in biology. Its various uses are thanks to its unique mathematical property: the rate of change of an exponential function is directly proportional to its current value.

 In other words, exponential functions serve as a kind of building block for modeling an array of phenomena—the aforementioned, plus the spread of fiseases and even natural processes, like radioactive decay and cooling.³

These exponential functions are but one of the simple building blocks of mathematics, starting with Algebra. With Savvas’s College Algebra course, students learn to tackle more than just exponential functions—through real-world examples, they’ll learn to break down complicated concepts into simple building blocks, covering concepts from exponential functions to graphing functions. 

Common Misconceptions of Exponential Functions

Equating exponential growth with rapid increase. Exponential growth doesn’t simplyreflect rapid increase. It also considers the underlying constant factor for change. 

Confusing exponential functions with polynomial ones. Whereas exponential functions have a positive constant as their base and a variable as their exponents, polynomial functions are functions of a single independent variable. 

Knowing how to write an exponential function and how to graph exponential functions can help clarify these differences.

Understanding Exponential Functions

Understanding exponential functions involves grasping the structure of constant change and the role of the exponent. 

By learning to convert multiplication into exponential form and practicing with real-world examples, students can accurately interpret, model, and communicate exponential changes in various disciplines. Knowing what an exponential function is and how to write an exponential function is the very first step.