An exponential duty is a mathematics function, i m sorry is used in countless real-world situations. That is largely used to discover the exponential decay or exponential expansion or to compute investments, version populations and so on. In this article, you will certainly learn around exponential duty formulas, rules, properties, graphs, derivatives, exponential series and examples.

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Table the 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 dubbed the base of the duty and it have to be greater than 0. The most typically used exponential duty base is the transcendental number e, i m sorry is roughly equal come 2.71828.

Exponential function Formula

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

The exponential role is an important mathematical role which is that the form

f(x) = ax

Where a>0 and also a is no equal come 1.

x is any real number.

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

Where r is the development percentage.

Exponential Decay

In Exponential Decay, the quantity decreases very rapidly in ~ first, and also then slowly. The price of adjust decreases over time. The rate of readjust becomes slow as time passes. The rapid growth meant to it is in an “exponential decrease”. The formula to specify 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 seen that as the exponent increases, the curves get steeper and also the rate of growth increases respectively. Thus, for x > 1, the value of y = fn(x) boosts for raising values the (n).


From the above, it have the right to be viewed that the nature that polynomial attributes is dependent on its degree. Higher the degree of any type of polynomial function, then higher is that is growth. A duty which grows quicker than a polynomial role is y = f(x) = ax, whereby a>1. Thus, for any type of of the confident integers n the function f (x) is claimed to grow quicker than the of fn(x).

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

It must be listed that exponential function is increasing and the point (0, 1) constantly lies on the graph of one exponential function. Also, the is very close to zero if the worth of x is largely negative.

Exponential role having base 10 is well-known as a usual exponential function. Think about the adhering to series:

The value of this series lies between 2 & 3. It is stood for by e. Maintaining e together base the function, we obtain y = ex, which is a an extremely important function in mathematics recognized as a organic exponential function.

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


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

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


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

Exponential duty Derivative

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

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

It is provided that the exponential duty f(x) =ex has a distinct property. It way that the derivative that the duty is the function itself.

(i.e) f ‘(x) = ex = f(x)

Exponential Series

The exponential collection are offered below.


Exponential function Properties

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

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

The graph of role y=2x is presented below. First, the residential or commercial property of the exponential duty graph as soon as the basic is greater than 1.


Exponential duty Graph for y=2x

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

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The domain is all real numbersThe variety is y>0The graph is increasingThe graph is asymptotic come the x-axis together x approaches an adverse infinityThe graph increases without bound as x approaches confident 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 duty and the graph as soon as the basic is in between 0 and 1 are given.

The line passes through the suggest (0,1)The domain includes all real numbersThe range is of y>0It creates a diminish graphThe heat in the graph over is asymptotic come the x-axis as x approaches optimistic infinityThe line increases without bound as x approaches an unfavorable infinityIt is a constant graphIt develops a smooth graph