Definition

A Functional Exponent is a function in the form of f(x) = a^x, where x is a variable and a is a constant, and it is known as the base of the function and should be greater than 0. Functional exponent is also called exponential function.

Forms of Functional Exponent

f(x) = a^x
f(x) = ba^x , where b is also a constant
f(x) = e^x
f(x) = pe^qx , where p is also a constant

Examples of Functional Exponent

The most common example of the functional exponent is e^x, where the value of e is 2.718. Some other examples are as follows:-

5e^15x
(1/2)^x
7(8)^-0.7x, etc.

Properties of Functional Exponent

a⁰ = 1
(a^m)ⁿ = a^{m x n}
a^m x aⁿ = a^{(m + n)}
a^m / aⁿ = a^{(m – n)}
(a x b)^m = a^m x b^m
(a / b)^m = a^m / b^m
a^-m = 1/a

Relation between Functional Exponent and Logarithmic Functions

If there is any functional exponent that is in the form of:-

p^x = b

Then, it can be written as:-

log _p b = x

So, the relation between Functional Exponent and Logarithmic Functions is:-

p^x = b => log _p b = x

Some Graphs of Functional Exponent

1. Graph of 5^x

Let, f(x) = 5^x . So, some values of f(x) will be-

So, the Graph of 5^x will be:-

2. Graph for (1/2)^x

Let f(x) = (1/2)^x. So, some values of f(x) will be-

values of (1/2)x in exponential functions

So, the Graph of (1/2)^x will be:-

graph of (1/2)x in exponential functions

3. Graph of e^x

Let f(x) = e^x. So, some values of f(x) will be-

Value of e = 2.718

So, the graphs of the e^x will be-

Derivative of Functional Exponent or Exponential Function

d/dx (e^x) = e^x
d/dx (a^x) = a^x log a

Integration of Functional Exponents

∫ e^x dx = e^x + C
∫ a^x dx = (a^x / log a) + C

Where C is a constant.

Series Expansion of e^x

So, What is a functional exponent? The answer is functional exponent is the same as the exponential functions that we see in Mathematics, it’s just that we are calling exponential functions a functional exponent. We have seen most of the required things anyone needs to know about functional exponents. We hope this article on the Exponetinial Function Helps you.

If you have any questions/doubts in mind, please use the comments below.

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Also Read:

What is a Functional Exponent?

Definition

Forms of Functional Exponent

Examples of Functional Exponent

Properties of Functional Exponent

Relation between Functional Exponent and Logarithmic Functions