Exponential Formula: Definition, Explanation

Exponentiation Functions and exponentiation formulas are widely used in Mathematics to perform complex calculations with large numbers. It also represents the rapid growth of a given dependent variable with a few independent variables. In general, the exponential function represents the high growth rate. Even in Physics and Chemistry, these functions are very useful and important. Moreover, it can be seen that as the exponent increases, the curves become steeper, respectively, and the growth rate increases. 

Key Terms: Exponentiation Function, Exponentiation Formulas, Variable, Exponents, Constant, Coefficient, Real Numbers, Independent Variables, Compound Interest

Also read: Isosceles Triangle Theorems

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What is the Exponential Function? 

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As the name of an exponential is defined, it involves an exponent. This exponent is diagrammatical employing a variable instead of a constant. On the opposite hand, its base is represented with constant worth rather than a variable. 

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

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 that are defined by an equation of the form are called exponential decay functions if the change factor b follows the inequality 0 < b < 1. 

Exponential Equation

Exponential capabilities are used to version the boom of populations, carbon date artifacts, assist coroners to decide the time of death, compute the investments and compound interest, decay fee of radioactive factors in addition to many different applications. 

The video below explains this:

Exponential Formula Detailed Video Explanation:

Read More: Difference Between Power And Exponent

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Formula of Exponential Function

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If a is a number such that a>0 and a ≠ 1 an exponential function is a function in the form, 

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

Where a is the base and x can be any real number 

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Some Basic Formulas of Exponential Function

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Many useful formulas are available for exponential mathematics functions. Some of these formulas are as follows:

Here x and y are variables and a, b, m, n are constants

• Adding the Exponents

\(x^a * x^b = x^{a+b}\)

• Subtracting the Exponents

\(\frac{x^a}{x^b} = x^{a-b}\)

• Getting Exponents of Exponents

\((x^a)b) = ((x^a)b)\)

• Expanding Exponents of Products

\((xy)^a = x^a * y^a\)

• Giving Value for Zero Exponent

X⁰ = 1

• Giving Value for Unit Exponent

\(x^1 = x\)

• Giving Value of Negative Exponent

\((x^-(^n) = \frac{1}{x^n}\)

• Giving Value of Fractional Exponent

\(x^{\frac{m}{n}} = \sqrt[n]{x^m}\)

Laws of Exponents

Read More: Powers and Negative Exponents

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Things to Remember 

• Exponentiation functions and exponentiation formula are very useful in Maths for the computation of complex computations with large numbers.
• The exponential function depicts the high growth rate in general. 
• In exponential function, the exponent is represented using a variable instead of a constant.
• The base in exponential function is represented with constant value rather than a variable.
• There are numerous useful formulas used for exponential functions which are directly used to get values of unknown variables.

Also read: Calculus Formula