An exponential duty is a math function, i beg your pardon is provided in many real-world situations. That is mostly used to uncover the exponential degeneration or exponential expansion or to compute investments, design populations and so on. In this article, you will learn around exponential role formulas, rules, properties, graphs, derivatives, exponential series and examples.

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Table that Contents:

What is Exponential Function?

An exponential duty is a Mathematical role in form f (x) = ax, whereby “x” is a variable and also “a” is a constant which is referred to as the basic of the role and it must be better than 0. The most commonly used exponential duty base is the transcendental number e, i m sorry is roughly equal to 2.71828.

Exponential function Formula

An exponential duty is identified by the formula f(x) = ax, where the input variable x occurs as an exponent. The exponential curve relies on the exponential duty and it relies on the worth of the x.

The exponential role is an essential mathematical role which is the the form

f(x) = ax

Where a>0 and also a is not equal come 1.

x is any type of real number.

If the variable is negative, the role is undefined because that -1 x

Where r is the expansion percentage.

Exponential Decay

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

y = a ( 1- r )x

Where r is the degeneration percentage.

Exponential function Graph

The following figure represents the graph of exponents of x. It deserve to be watched that together the exponent increases, the curves gain steeper and the rate of growth increases respectively. Thus, for x > 1, the value of y = fn(x) increases for enhancing values of (n).


From the above, it have the right to be checked out that the nature of polynomial attributes is dependence on that is degree. Higher the degree of any kind of polynomial function, then greater is that is growth. A role which grows faster than a polynomial function is y = f(x) = ax, where a>1. Thus, for any kind of of the hopeful integers n the role f (x) is claimed to grow faster than that of fn(x).

Thus, the exponential duty having base greater than 1, i.e., a > 1 is characterized as y = f(x) = ax. The domain the exponential function will be the collection of whole real numbers R and the variety are said to it is in the set of every the hopeful real numbers.

It must be listed that exponential function is increasing and also the allude (0, 1) always lies top top the graph of one exponential function. Also, that is an extremely close come zero if the value of x is greatly negative.

Exponential function having base 10 is well-known as a typical exponential function. Consider the complying with series:

The worth of this collection lies between 2 & 3. The is represented by e. Keeping e as base the function, we acquire y = ex, which is a really important role in mathematics known as a organic exponential function.

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


For base a = 10, this function is known as usual logarithm and for the basic a = e, that is well-known as herbal logarithm denoted by ln x. Following are some of the important observations about logarithmic features which have a base a>1.

For the log function, though the domain is only the collection of optimistic real numbers, the variety is collection of all genuine values, i.e. RWhen us plot the graph of log in functions and also move from left to right, the functions show increasing behaviour.The graph the log role never cut x-axis or y-axis, though it appears to often tend towards them.


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

Exponential function Derivative

Let us now emphasis on the derivative the exponential functions.

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

It is detailed that the exponential duty f(x) =ex has actually a distinct property. It means that the derivative the the role is the role itself.

(i.e) f ‘(x) = ex = f(x)

Exponential Series

The exponential collection are offered below.


Exponential duty Properties

The exponential graph that a role represents the exponential function properties.

Let us consider the exponential function, y=2x

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


Exponential duty Graph for y=2x

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

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The domain is all real numbersThe selection is y>0The graph is increasingThe graph is asymptotic to the x-axis together x approaches an adverse infinityThe graph rises without bound together x approaches positive infinityThe graph is continuousThe graph is smooth


Exponential function Graph y=2-x 

The graph of duty y=2-x is presented above. The properties of the exponential duty and its graph when the basic is between 0 and 1 are given.

The line passes through the allude (0,1)The domain consists of all genuine numbersThe selection is that y>0It creates a to decrease graphThe line in the graph over is asymptotic come the x-axis together x approaches positive infinityThe line rises without bound together x approaches an unfavorable infinityIt is a continuous graphIt creates a smooth graph