Exponentiation Formula

Exponentiation functions and exponentiation formula are very much used in mathematics for doing complex computations with large numbers. It also represents the fast growth of a given dependent variable with some independent variables. Generally, the exponential function represents the high growth rate.  In physics as well in chemistry such functions are very useful and important. Also, it can be seen as the exponent increases, the curves get steeper and the rate of growth increases respectively. In this article, we will discuss the exponentiation formula with examples. Let us begin learning!

Exponentiation Formula

What is the Exponentiation Function?

As the name of an exponential function is defined, it involves an exponent. This exponent is represented using a variable rather than a constant. On the other hand, its base is represented with constant value rather than a variable.

Let \(f(x)=ab^x\)

This is an exponential function where “b” is a constant, the exponent “x” is the independent variable i.e. input of the function. The coefficient “a” is called the initial value of the function, f(x) represents the dependent variable i.e. output of the function. Thus for x > 1, the value of f(x) will always increase for increasing values of x.

The exponential property can be used to solve the equations with exponential functions. Exponential functions defined by an equation of the form as above are called exponential decay functions if the change factor b follows the inequality 0 < b < 1.

The good thing about exponential functions is that they are very useful in real-world situations. Exponential functions are used to model the growth of populations, carbon date artifacts, help coroners to determine the time of death, compute the investments and compound interest, decay rate of radioactive elements as well as many other applications.

Some Basic Exponential Formula:

In mathematics many useful formulas are available for exponential functions. We can directly use these in various equations to get values of unknown variables. Some of these formulas are given bellow:

Here we have assumed that x and y are variables and a, b, m, n are constants.

(1) \(x^a\times x^b =x^{a+b}\) this is adding the exponets.

(2) \(\frac{x^a}{x^b} = x^{a-b}\)   this is subtracting the exponents.

(3) \((x^a)^b) = ((x^a)^b)\)  this is getting exponents of exponents.

(4) \((xy)^a = x^a\times y^a\)  this is expanding exponents of products.

(5)\(x^0 = 1\) this is giving value for zero exponent.

(6) \(x^1 = x\)  this is giving value for unit exponent.

(7) \((x^-(^n) = \frac{1}{x^n}\) this is giving value of negative exponent.

(8)\( x^\frac{m}{n}= \sqrt[n]{x^m}\) this is giving value of fractional exponent.

Solved Examples

Q.1: Find value of x  , \(4^{4x-5}= 16^2\)

Solution:

Given: \(4^{4x-5}= 16^2\)

We can express 16 as a power of 4,

i.e. \(4^{4x-5}= {4^2}^2\)

i.e. \(4^{4x-5} = 4^4\)

Now comparing both sides of the above equation.

4x-5 = 4

4x = 4+5

4x = 9

x = \(\frac{9}{4}\)

Q.2: Given the function f(x)=4^x  , then evaluate each of the following.

(i) f(-2)

(ii) f(1)

(iii) f(0)

Solution:

(i) \(f(x) = 4^x\)

So,  put x=-2 we get

\(f(-2)=4^{-2}\)

i.e. \(f(-2) = \frac{1}{4^2}\)

i.e. \(f(-2) = \frac{1}{16}\)

(ii) \(f(x) = 4^x\)

Put x = 1

\(f(1)  = 4^1\)

i.e. \(f(1) = 4\)

(iii) \(f(x) = 4^x\)

put x=0

we get \(f(0)= 4^0\)

i.e. f(0) = 1