An exponential role is a math function, i beg your pardon is used in countless real-world situations. That is largely used to uncover the exponential degeneration or exponential development or come compute investments, model populations and also so on. In this article, you will certainly learn around exponential function formulas, rules, properties, graphs, derivatives, exponential collection and examples.

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Table of Contents:

What is Exponential Function?

An exponential duty is a Mathematical duty in type f (x) = ax, whereby “x” is a variable and “a” is a constant which is called the basic of the duty and it must be greater than 0. The most frequently used exponential function base is the transcendental number e, i beg your pardon is about equal to 2.71828.

Exponential duty Formula

An exponential duty is characterized by the formula f(x) = ax, where the input change x occurs together an exponent. The exponential curve depends on the exponential duty and it counts on the worth of the x.

The exponential duty is an important mathematical duty 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 very rapidly at first, and also then slowly. The rate of adjust decreases end time. The price of readjust becomes slower as time passes. The rapid expansion 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 index number of x. It can be viewed that together the exponent increases, the curves gain steeper and also the price of expansion increases respectively. Thus, because that x > 1, the worth of y = fn(x) rises for enhancing values that (n).


From the above, it can be seen that the nature that polynomial attributes is dependent on that is degree. Higher the level of any type of polynomial function, then greater is that growth. A role which grows much faster than a polynomial role is y = f(x) = ax, wherein a>1. Thus, for any of the optimistic integers n the duty f (x) is claimed to grow faster than that of fn(x).

Thus, the exponential duty having base better than 1, i.e., a > 1 is defined as y = f(x) = ax. The domain of exponential role will be the collection of entire real numbers R and the variety are said to be the collection of every the positive real numbers.

It have to be detailed that exponential role is increasing and also the allude (0, 1) always lies ~ above the graph of an exponential function. Also, that is really close to zero if the value of x is mostly negative.

Exponential function having base 10 is recognized as a typical exponential function. Consider the following series:

The value of this collection lies between 2 & 3. That is stood for by e. Keeping e as base the function, we gain y = ex, which is a really important function 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 well-known as logarithmic function.


For basic a = 10, this role is well-known as common logarithm and for the basic a = e, the is known as natural logarithm denoted by ln x. Following are several of the vital observations about logarithmic functions which have a base a>1.

For the log function, despite the domain is only the set of optimistic real numbers, the selection is collection of all genuine values, i.e. RWhen us plot the graph of log functions and also move indigenous left come right, the functions display increasing behaviour.The graph that log function 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 bµ = aLogbpq = Logbp + LogbqLogbpy = ylogbpLogb (p/q) = logbp – logbq

Exponential duty Derivative

Let us now emphasis on the derivative of exponential functions.

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

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

(i.e) f ‘(x) = ex = f(x)

Exponential Series

The exponential collection are given below.


Exponential duty Properties

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

Let us take into consideration the exponential function, y=2x

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


Exponential function Graph for y=2x

The graph passes v the allude (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 boosts without bound as x approaches positive infinityThe graph is continuousThe graph is smooth


Exponential role Graph y=2-x 

The graph of duty y=2-x is shown above. The nature of the exponential function and the graph as soon as the basic is between 0 and also 1 are given.

The heat passes v the allude (0,1)The domain contains all genuine numbersThe selection is the y>0It develops a diminish graphThe heat in the graph above is asymptotic to the x-axis together x approaches positive infinityThe line boosts without bound together x approaches negative infinityIt is a constant graphIt develops a smooth graph