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Algebra / Exponential & Logarithmic Functions

Exponential Functions

An exponential function puts the variable in the exponent. That one change — 2ˣ instead of x² — produces a function that multiplies by the same factor at every step, which is why it describes compound interest, population growth and radioactive decay.

Practice Problems
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Chapter 6 · Lesson 1 of 15Exponential & Logarithmic Functions
  1. 1.Exponential Functions
  2. 2.Graphing Exponential Functions
  3. 3.Writing Exponential Functions
  4. 4.Exponential Growth and Decay
  5. 5.Logarithms
  6. 6.Common and Natural Logarithms
  7. 7.Logarithm Rules
  8. 8.Change of Base Formula
  9. 9.Graphing Logarithmic Functions
  10. 10.Exponential vs Logarithmic Functions
  11. 11.Solving Exponential Equations
  12. 12.Solving Logarithmic Equations
  13. 13.Continuous Compound Interest
  14. 14.Exponential vs Logistic Growth
  15. 15.Richter Scale and Earthquake Intensity
See all algebra chapters →

Most functions you have met so far — lines, parabolas, polynomials — raise the variable to a fixed power. An exponential function turns that around and raises a fixed number to a variable power. The result grows (or shrinks) by the same factor every time increases by one, and that repeated multiplication is the mathematics behind savings accounts, bacteria cultures, drug half-lives and the logarithms that come later in this chapter.

Exponential function in standard form f(x) = a times b to the x, with b greater than 0 and b not equal to 1
Standard form of an exponential function

What Is an Exponential Function?

An exponential function is a function of the form

The input is the exponent. The base is a fixed positive number. Every time goes up by , the output is multiplied by again.

The simplest example is :

Each step to the right doubles the output; each step to the left halves it. The negative inputs use the negative exponent rule, , so the output never reaches zero and never goes negative.

The Parts of an Exponential Function

In the standard form , each letter has a job:

PartNameWhat it tells you
initial valuethe output at , since gives — the y-intercept
base, or growth factorthe number the output is multiplied by for each step in
exponentthe input — how many times the factor has been applied

What is b in an exponential function? It is the constant multiplier. If , the function grows (exponential growth). If , it shrinks toward zero (exponential decay). The growth and decay lesson builds the real-world models from this one number.

Graphs of y = 2^x rising and y = (1/2)^x falling, both passing through (0, 1)
b > 1 grows, 0 < b < 1 decays — both pass through (0, a)

Why the Base Has Restrictions

The conditions and are not arbitrary:

  • : for every input, so is just the constant . Nothing grows or decays.
  • : is fine at whole numbers but undefined at , because is not real. The graph would be a scatter of dots, not a curve.
  • : is for positive and undefined for negative .

A positive base other than gives a smooth curve defined for every real number, and that is what makes it a genuine exponential function.

How to Tell If a Function Is Exponential

Two quick tests settle whether a function is exponential.

From the formula: the variable must be in the exponent, with a constant positive base that is not .

From a table: equal steps in must multiply by the same factor. Divide each output by the previous one — if the ratio is constant, the data is exponential.

Example 1: Which of the Following Are Exponential Functions?

FunctionExponential?Reason
yesconstant base , variable exponent
novariable base, fixed exponent — a power function
yes, — exponential decay
nonegative base
nobase gives the constant
yes, base
yes, shiftedan exponential moved up units

The two that trip people up are , which looks exponential but has the variable in the wrong place, and , which is exponential decay in disguise.

Evaluating Exponential Functions

Evaluating an exponential function means substituting the input into the exponent and simplifying. Apply the exponent first, then multiply by — the exponent belongs to alone, not to .

Example 2: Evaluation of Exponential Functions

Take and a few related functions:

ExpressionWorkResult
for
for
for

Fractional inputs use rational exponents: the denominator is a root, the numerator a power.

The Natural Exponential Function

One base appears far more than any other: Euler’s number,

The natural exponential function is . Like , is irrational, so you evaluate it with the key on a calculator:

ExpressionValue (to 4 decimal places)

Since , is a growth function. It sits between and on a graph, and its inverse — the natural logarithm — is covered in the common and natural logarithms lesson.

Linear and Exponential Functions Compared

The cleanest way to see what makes a function exponential is to set it next to a linear one.

Linear
Exponential
  • A linear function adds the same amount at each step (here ). Its rate of change is constant.
  • An exponential function multiplies by the same factor at each step (here ). Its rate of change keeps increasing.

Early on the two are close — the linear one is even ahead at — but an exponential with eventually overtakes any linear function, no matter how steep the line.

Line y = 3x + 1 and curve y = 3^x, with the exponential overtaking the line after x = 1
Adding 3 each step versus multiplying by 3 each step

Worked Example

A table gives , , , . Decide whether it is linear or exponential, write the function, and find .

Step 1 — check the differences:

The differences are not constant, so it is not linear.

Step 2 — check the ratios:

A constant ratio means the data is exponential with .

Step 3 — read the initial value at :

Step 4 — write the function:

Step 5 — evaluate at :

Common Mistakes to Avoid

  • Multiplying before applying the exponent. is , not . The exponent attaches to the base only.
  • Confusing with . At , but . Check where the variable is before calling a function exponential.
  • Thinking a negative exponent makes the output negative. , which is positive. The output of with is always positive.
  • Using differences instead of ratios on a table. Exponential data has constant ratios; constant differences mean the data is linear.

Practice Problems

Work each one before opening the answer.

Problem 1. For , find and .

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Step 1 — apply the exponent, then multiply:

Step 2 — a negative exponent is a reciprocal:

Answer: ,

Problem 2. Evaluate at .

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Step 1 — compute the power:

Step 2 — multiply by the initial value:

Answer:

Problem 3. Which of these are exponential functions? , , , .

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Step 1 — : constant base , variable exponent — exponential.

Step 2 — : the variable is the base — a power function.

Step 3 — : , with — exponential decay.

Step 4 — : negative base — not an exponential function.

Answer: and

Problem 4. For , find and .

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Step 1 — a negative exponent flips the fraction:

Step 2 — write as , take the square root, then cube:

Answer: ,

Problem 5. Evaluate at , to three decimal places.

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Step 1 — simplify the exponent:

Step 2 — evaluate and multiply:

Answer:

Problem 6. Is the data , , , linear or exponential?

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Step 1 — check the differences:

Step 2 — check the ratios for comparison:

The differences are constant and the ratios are not.

Answer: linear,

With the definition, the parts and the evaluation rule in place, the next step is to see the whole curve: the graphing exponential functions lesson covers the shape, the asymptote, domain and range, and how transformations move the graph.

Frequently Asked Questions

Why can't the base of an exponential function be negative or 1?+

A base of 1 gives for every , which is a constant function, not an exponential one. A negative base breaks down for fractional inputs: is not a real number, so the function would have holes at most values of . Requiring and avoids both problems.

What is the difference between an exponential function and a power function?+

It is where the variable sits. In the power function the base varies and the exponent is fixed; in the exponential function the base is fixed and the exponent varies. The difference in growth is enormous: at , while .

What is the natural exponential function?+

It is , the exponential function whose base is Euler's number . It is the only exponential whose rate of growth at every point equals its own value, which is why it appears throughout calculus, continuous compounding and science formulas.

Related lessons

Exponent RulesRational ExponentsFunctions: The One-Output Rule and the Vertical Line TestGraphing Exponential Functions