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# Exponential Function

Top
 Sub Topics In mathematics, we have dealt with various such functions as x = y$^3$, where base was the variable and power was constant. But in the case of exponentials, however, we have to deal with functions where base is the fxed number, and the power is the variable for example: x = 4$^y$, y = 2$^{2x}$ and f(x) = 10$^x$ are exponential functions. Also natural exponential function can be represented as y = e$^x$ or exp(x), where e is Euler's number and its value is approximately 2.718.  Exponential functions are commonly used to calculate exponential decay and exponential growth. Here will study about exponential function in detail.

## Definition

A exponential function is a mathematical expression where a variable represents the exponent of an expression. Also it's the inverse of a logarithm function.
If a is any number a $\neq$ 1 and a > 0 then an exponential function is in the form, y = f(x) = a$^x$
where x is any real number and a is the base.

## Rules

Following are some important rules of exponential functions:

1) $z^0$ = 1

2)
$z^n$ = z.z.z......n times

3)
$z^{-n}$ = $\frac{1}{z^n}$

4)
$z ^{\frac{p}{q}}$ = $\sqrt[q]{z^p}$

5)
$z^p \times z^q = z^{p+q}$

6)
$(zs)^p=z^ps^p$

7)
$(\frac{z}{s})^n$=$\frac{z^n}{s^n}$

8)
$\frac{z^p}{z^q}$ = $z^{p-q}$

9)
$(z^p)^q=z^{pq}$

10)
If $z^m=s^n$ then m = n.

## Graph

There are certain characteristics need to follow while graphing functions of the form y = a$^x$:

1) Graph always decrease when "a" lies between 0 and 1.

2) Graph increase when a > 1.

3) Range is all positive real numbers and domain of the function is all real numbers.

4) When graph passes the vertical line test it shows graph is a function and when graph passes the horizontal line test, it shows its inverse is also a function.

Let us draw a graph for functions y = 2$^x$ and y = 8$^x$.

## Examples

Some examples on exponential functions are given below:

Example 1: Solve for y when x = 4

y = 2(5$^x$)

Solution:  y = 2(5$^x$)

Plug in the value, x = 4 in above equation

y = 2(5$^4$) = 2(5 . 5. 5. 5) = 1250

Example 2: Evaluate $\frac{16a^4}{2a^2}$

Solution: $\frac{16a^4}{2a^2}$ = $\frac{2^4a^4}{2a^2}$

= $2^{4-1} \times a^{4-2}$ (using exponential rules)

= $2^3a^2$

Example 3:  Find the value of a. If $4^a$ - 1 = 63.

Solution: $4^a$ - 1 = 63

$4^a$ - 1 + 1 = 63 + 1

$4^a$ = 64

$4^a$ = $4^3$

a = 3 (Because If $z^m=s^n$ then m = n)

The value of a is 3.