Exponential Functions: Definition, Formula and Examples

Exponential Function is a Mathematical function that involves exponents. It is written in the form f (x) = a^x, where “x” is a variable and “a” is a constant. The constant a is the base of the function and it should be greater than 0.

An exponential function is classified into two types, i.e., exponential growth and exponential decay. If the function is increasing then it’s called exponential growth and if the function is decreasing then it’s called exponential decay.

Let’s know more about Exponential Function definition, formula, properties and examples in detail below.

An exponential function is a mathematical function of the form f(x) = a.b^x, where:

• a is a constant term known as the coefficient
• b is the base of the exponential function, which is a positive constant,
• x represents the exponent, which can be any real number.

Exponential Function Definition

A mathematical function in the form of f(x) = a^x, where “a” is called the base of the function, which is a constant greater than 0, and “x” is the exponent of the function, which is a variable. When x > 1, the function f(x) increases with increasing x values. The value of “e” is approximately equal to 2.71828.

The formula of exponential function is given as follows:

f(x) = a^x, where a>0 and a ≠ 1 and x ∈ R

The exponential function is classified into two types based on the growth or decay of an exponential curve. They are :

1. Exponential Growth
2. Exponential Decay

Exponential Growth

In Exponential Growth, a quantity increases slowly and then progresses rapidly. An exponentially growing function has an increasing graph. It can be used to illustrate economic growth, population expansion, compound interest, growth of bacteria in a culture, population increases, etc.

The formula for exponential growth is:

y = a(1 + r)^x

where, r is Growth Percentage

Exponential Decay

In Exponential Decay, a quantity decreases very rapidly at first and then fades gradually. An exponentially decaying function has a decreasing graph. The concept of exponential decay can be applied to determine half-life, mean lifetime, population decay, radioactive decay, etc.

The formula for exponential decay is:

y = a(1 – r)^x

where, r is Decay Percentage

The image given below represents the graphs of the exponential functions y = e^x and y = e^-x. From the graphs, we can understand that the e^x graph is increasing while the graph of e^-x is decreasing. The domain of both functions is the set of all real numbers, while the range is the set of all positive real numbers.

For an exponential function y = a^x (a>1), the logarithm of y to base e is x = log_ay, which is the logarithmic function. Now, observe the graph of the natural logarithmic function y = log_ex. From the graph, we can notice that a logarithmic function is only defined for positive real values.

As the logarithmic function is not defined for negative values, its domain is the set of all positive real numbers.

Exponential function e^x can be expressed as an infinite series. This series is known as its Taylor series expansion, and it is derived using the function’s derivatives evaluated at zero.

Exponential function e^x is defined as:

e^x = ∑ xⁿ/n!

where n! (n factorial) is the product of all positive integers up to n.

Derivation of Exponential Function Series

Consider the exponential function f(x) = e^x

We know that,

d/dx (e^x) = d²/dx²(e^x) = … = dⁿ/dxⁿ(e^x) = e^x

Taylor series expansion of a function f(x) about x = 0 is given by:

f(x) = f(0) + f′(0).x/1! + f′′(0).x²/2! + f′′′(0).x³/3! + ⋯ …(i)

f(x) = ∑ {f⁽ⁿ⁾(0)​/n!}xⁿ

f(x) = e^x, we have:

f(0) = e⁰ = 1

f′(0) = e⁰ = 1

f′′(0) = e⁰ = 1

f′′′(0) = e⁰ = 1

…

All derivatives of e^x evaluated at x = 0 are equal to 1.

Substituting these values into the Taylor series formula eq(i)

e^x = 1 + x/1!​ + x²/2!​ + x³/3! ​+ ⋯

Thus, exponential function series is derived.

Below are the some properties of Exponential Function.

Domain and Range

• Domain: The domain of an exponential function is all real numbers, (−∞,∞)
• Range: The range depends on the value of a. If a>0, the range is (0,∞). If a<0, the range is (−∞,0).

Intercepts

• Y-intercept: When x=0, f(0)=a⋅b⁰=a. Thus, the y-intercept is at (0,a).
• X-intercept: For the standard exponential function a⋅b^x, there is no x-intercept because b^x≠0 for any real x.

Asymptotes

• Exponential functions have a horizontal asymptote. For f(x)=a⋅b^x, if a>0, the horizontal asymptote is y=0. If a<0, the horizontal asymptote is also y=0, but the function approaches it from below.

For a typical exponential function of the form, f(x) = ax, where a is a positive constant, the domain encompasses all real numbers. This means that you can input any real number x into the function.

On the other hand, the range of an exponential function is limited to positive real numbers. No matter what real number you choose for x, the output of f(x) will always be greater than zero. This is because any positive number raised to a power, whether that power is positive, negative, or zero, will result in a positive number.

Thus, the range of f(x) = a^x is (0, ∞), indicating that the function never touches or crosses the x-axis but grows indefinitely as x increases.

Exponential Graph of f(x) = 2^x

Let us consider an exponential function f(x) = 2^x.

From the graph, we can observe that the graph of f(x) = 2^x is upward-sloping, increasing faster as the value of x increases. The graph formed is increasing and is also smooth and continuous. The graph lies above the X-axis and passes through (0, 1).

As x approaches negative infinity, the graph becomes arbitrarily close to the X-axis. The domain of an exponential function holds all real values, whereas its range contains all values greater than zero (y>0).

Following are some of the important formulas used for solving problems involving exponential functions:

Article Related to Exponential Functions:

• Exponential Function Formula
• Derivative of Exponential Function
• Exponential Graph

Let’s solve some questions on the Exponential Functions.

Example 1: Simplify the exponential function 5^x – 5^x+3.

Solution:

Given exponential function: 5^x – 5^x+3

From the properties of an exponential function, we have a^x × a^y = a^{(x + y)}

So, 5^x+3 = 5^x × 5³ = 125×5^x

Now, the given function can be written as

5^x – 5^x+3 = 5^x – 125×5^x

= 5^x(1 – 125)

=5^x(–124)

= –124(5^x)

Hence, the simplified form of the given exponential function is –124(5^x).

Example 2: Find the value of x in the given expression: 4³× (4)^x+5 = (4)^2x+12.

Solution:

Given,

4³× (4)^x+5 = (4)^2x+12

From the properties of an exponential function, we have a^x × a^y = a^{(x + y)}

⇒ (4)^3+x+5 = (4)^2x+12

⇒(4)^x+8 = (4)^2x+12

Now, as the bases are equal, equate the powers.

⇒ x+8 = 2x+12

⇒ x – 2x = 12 – 8

⇒ – x = 4

⇒ x = –4

Hence, the value of x is –4.

Example 3: Simplify: (3/4)^–6 × (3/4)⁸.

Solution:

Given: (3/4)^–6 × (3/4)⁸

From the properties of an exponential function, we have a^x × a^y = a^{(x + y)}

Thus, (3/4)^–6×(3/4)⁸ = (3/4)^(–6+8)

= (3/4)²

= 3/4 × 3/4 = 9/16

Hence, (3/4)^–6 × (3/4)⁸ = 9/16.

Example 4: In the year 2009, the population of the town was 60,000. If the population is increasing every year by 7%, then what will be the population of the town after 5 years?

Solution:

Given data:

• Population of the town in 2009 (a) = 60,000
• Rate of increase (r) = 7%
• Time span (x) = 5 years

Now, by the formula for the exponential growth, we get,

y = a(1+ r)^x

= 60,000(1 + 0.07)⁵

= 60,000(1.07)⁵

= 84,153.1038 ≈ 84,153.

So, the population of the town after 5 years will be 84,153.

Q1. Calculate the value of f(x) for f(x) = 3.2^x when x = 4.

Q2. Given the exponential function g(x) = 5.(0.5)^x, sketch the graph of the function. Indicate the behavior of the function as x increases and as x decreases. Identify any asymptotes and intercepts.

Q3. A population of bacteria doubles every hour. If the initial population is 200 bacteria, express the population P as an exponential function of time t in hours. Then, find the population after 6 hours.

Q4. Solve for x in the exponential equation 10.3^x = 90.

What is Exponential Function?

An exponential function is defined as a mathematical function with the formula f(x) = ax, where “x” is a variable and is known as the exponent of the function, and “a” is a constant greater than zero and is known as the base of the function.

What are Properties of Exponential Function?

Some properties of Exponential Function are:

• Domain of an exponential function holds all real values, whereas its range contains all values greater than zero (y>0).
• Its graph is upward-sloping, increasing faster as the value of x increases. The graph lies above the X-axis and passes through (0, 1). As x approaches negative infinity, the graph becomes arbitrarily close to the X-axis.

What are Exponential Formulas?

The following are some exponential formulas for exponential functions. They hold true if a > 0 and for all real values of m and n.

• a^x × a^y = a^{(x + y)}
• a^x/ay = a^{(x – y)}
• a⁰ = 1
• a^-x = 1/a^x
• (a^m)ⁿ = a^mn

What are Types of Exponential Functions?

Exponential function is classified into two types based on the growth or decay of an exponential curve, i.e., exponential growth and exponential decay.

What are Formula of Exponential Growth and Exponential Decay?

Formula for exponential growth is:

y = a(1+ r)^x where r is the growth percentage

Formula for exponential decay is:

y = a(1 – r)^x, where r is the decay percentage