An exponential function is a mathematical function, i m sorry is used in countless real-world situations. It is largely used to find the exponential decay or exponential expansion or come compute investments, model populations and so on. In this article, you will certainly learn about exponential function formulas, rules, properties, graphs, derivatives, exponential collection and examples.

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Table the Contents:

What is Exponential Function?

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

Exponential function Formula

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

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

f(x) = ax

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

x is any type of real number.

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

Where r is the expansion percentage.

Exponential Decay

In Exponential Decay, the quantity decreases very rapidly in ~ first, and also then slowly. The rate of adjust decreases end time. The price of readjust becomes slower as time passes. The rapid growth meant to be an “exponential decrease”. The formula to define the exponential expansion is:

y = a ( 1- r )x

Where r is the decay percentage.

Exponential role Graph

The following number represents the graph of index number of x. It have the right to be checked out that as the exponent increases, the curves get steeper and also the price of growth increases respectively. Thus, because that x > 1, the worth of y = fn(x) boosts for raising values that (n).


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

Thus, the exponential role having base greater than 1, i.e., a > 1 is defined as y = f(x) = ax. The domain that exponential role will be the collection of whole real number R and also the selection are stated to it is in the collection of every the confident real numbers.

It have to be listed that exponential function is increasing and also the allude (0, 1) constantly lies on the graph of an exponential function. Also, the is an extremely close to zero if the value of x is mostly negative.

Exponential duty having base 10 is well-known as a common exponential function. Take into consideration the following series:

The worth of this series lies in between 2 & 3. It is represented by e. Keeping e as base the function, we obtain y = ex, i m sorry is a really important duty in mathematics recognized as a herbal 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 role is well-known as logarithmic function.


For basic a = 10, this role is known as common logarithm and also for the basic a = e, that is known as natural logarithm denoted by ln x. Complying with are some of the necessary observations about logarithmic attributes which have a base a>1.

For the log in function, despite the domain is just the collection of positive real numbers, the selection is collection of all actual values, i.e. RWhen we plot the graph of log in functions and also move indigenous left to right, the functions present increasing behaviour.The graph the log role never cuts x-axis or y-axis, despite it seems to have tendency towards them.


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

Exponential function Derivative

Let united state now focus on the derivative that exponential functions.

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

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

(i.e) f ‘(x) = ex = f(x)

Exponential Series

The exponential series are offered below.


Exponential role Properties

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

Let us think about the exponential function, y=2x

The graph of duty y=2x is displayed below. First, the property of the exponential role graph once the base is higher than 1.


Exponential function Graph for y=2x

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

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The domain is all genuine numbersThe selection is y>0The graph is increasingThe graph is asymptotic come the x-axis together x approaches an adverse 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 role y=2-x is shown above. The properties of the exponential duty and that graph as soon as the base is between 0 and 1 are given.

The heat passes with the suggest (0,1)The domain consists of all genuine numbersThe variety is the y>0It forms a diminish graphThe heat in the graph over is asymptotic come the x-axis together x approaches confident infinityThe line rises without bound together x approaches an unfavorable infinityIt is a continuous graphIt creates a smooth graph