An exponential function is a mathematical function, which is used in many real-world situations. It is mainly used to find the exponential decay or exponential growth or to compute investments, model populations and so on. In this article, you will learn about exponential function formulas, rules, properties, graphs, derivatives, exponential series and examples.

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Table of Contents:

What is Exponential Function?

An exponential function is a Mathematical function in form f (x) = ax, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0. The most commonly used exponential function base is the transcendental number e, which is approximately equal to 2.71828.

Exponential Function Formula

An exponential function is defined by the formula f(x) = ax, where the input variable x occurs as an exponent. The exponential curve depends on the exponential function and it depends on the value of the x.

The exponential function is an important mathematical function which is of the form

f(x) = ax

Where a>0 and a is not equal to 1.

x is any real number.

If the variable is negative, the function is undefined for -1 x

Where r is the growth percentage.

Exponential Decay

In Exponential Decay, the quantity decreases very rapidly at first, and then slowly. The rate of change decreases over time. The rate of change becomes slower as time passes. The rapid growth meant to be an “exponential decrease”. The formula to define the exponential growth is:

y = a ( 1- r )x

Where r is the decay percentage.

Exponential Function Graph

The following figure represents the graph of exponents of x. It can be seen that as the exponent increases, the curves get steeper and the rate of growth increases respectively. Thus, for x > 1, the value of y = fn(x) increases for increasing values of (n).


From the above, it can be seen that the nature of polynomial functions is dependent on its degree. Higher the degree of any polynomial function, then higher is its growth. A function which grows faster than a polynomial function is y = f(x) = ax, where a>1. Thus, for any of the positive integers n the function f (x) is said to grow faster than that of fn(x).

Thus, the exponential function having base greater than 1, i.e., a > 1 is defined as y = f(x) = ax. The domain of exponential function will be the set of entire real numbers R and the range are said to be the set of all the positive real numbers.

It must be noted that exponential function is increasing and the point (0, 1) always lies on the graph of an exponential function. Also, it is very close to zero if the value of x is mostly negative.

Exponential function having base 10 is known as a common exponential function. Consider the following series:

The value of this series lies between 2 & 3. It is represented by e. Keeping e as base the function, we get y = ex, which is a very important function in mathematics known as a natural exponential function.

For a > 1, the logarithm of b to base a is x if ax = b. Thus, loga b = x if ax = b. This function is known as logarithmic function.


For base a = 10, this function is known as common logarithm and for the base a = e, it is known as natural logarithm denoted by ln x. Following are some of the important observations regarding logarithmic functions which have a base a>1.

For the log function, though the domain is only the set of positive real numbers, the range is set of all real values, i.e. RWhen we plot the graph of log functions and move from left to right, the functions show increasing behaviour.The graph of log function never cuts x-axis or y-axis, though it seems to tend towards them.


Logap = α, logbp = β and logba = µ, then aα = p, bβ = p and bµ = aLogbpq = Logbp + LogbqLogbpy = ylogbpLogb (p/q) = logbp – logbq

Exponential Function Derivative

Let us now focus on the derivative of exponential functions.

The derivative of ex with respect to x is ex, i.e. d(ex)/dx = ex

It is noted that the exponential function f(x) =ex has a special property. It means that the derivative of the function is the function itself.

(i.e) f ‘(x) = ex = f(x)

Exponential Series

The exponential series are given below.


Exponential Function Properties

The exponential graph of a function represents the exponential function properties.

Let us consider the exponential function, y=2x

The graph of function y=2x is shown below. First, the property of the exponential function graph when the base is greater than 1.


Exponential Function Graph for y=2x

The graph passes through the point (0,1).

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The domain is all real numbersThe range is y>0The graph is increasingThe graph is asymptotic to the x-axis as x approaches negative infinityThe graph increases without bound as x approaches positive infinityThe graph is continuousThe graph is smooth


Exponential Function Graph y=2-x 

The graph of function y=2-x is shown above. The properties of the exponential function and its graph when the base is between 0 and 1 are given.

The line passes through the point (0,1)The domain includes all real numbersThe range is of y>0It forms a decreasing graphThe line in the graph above is asymptotic to the x-axis as x approaches positive infinityThe line increases without bound as x approaches negative infinityIt is a continuous graphIt forms a smooth graph