exponential functions

An exponential function is a unique mathematical function of mathematics. It is denoted as f(x)=ex, where X is the variable and e is the constant written as an exponent. There are other exponential functions also, They are characterized the same as this function. When continuous growth or continuous decay in the quantity of a particular set of proportions are observed, The exponential function arises. Exponential functions hold an important chunk in the syllabus of XI-XII standard students. Understanding the concept of exponential functions is very crucial for students.

These functions are used widely in many mathematical calculations and also are part of the school syllabus of the exams of XI and XII. These functions are also used in real-world situations, showing the importance of these functions.

About Exponential functions

Exponential functions are used to calculate real-world observations, and are used in numerical of various subjects like physics, mathematics, stats, economics, etc. to obtain results. Exponential functions are inverse of logarithmic functions, exponential functions have their laws and properties to solve the exponential problems.

In brief

The exponential function is the function that grows faster in both consequences i.e. growth or decay. It is in the form of f(x) =abx, where b > 1. This function is performed to obtain the position i.e., growing or decaying of functions that are in observation. 

Formula and results

Exponential function formula: f(x)=ax. First, let’s understand the terms of the formula of exponential functions. In f(x) = ax, f(x)can be described as a function of X, and  X is exponent and a is any real value, Sometimes a is constant that is used to run this function. Where x is any real number, then the value of the exponent can be any greater than 0 and not equal to one. Exponent at 1 is always carried out, called a base. The value of exponential (e) at 1 is 2.71828 which is also termed ‘Euler’s number’.

If x is a negative observation, the value of the function remains unspecified and lies between 0 to 1 (0 < x < 1). This formula when represented on a graph forms a curve in a continuous direction, i.e., upward or downward. If the curve is directed upward, then it is exponential growth, and if downward it is said as exponential decay. Exponential growth and decay are the main points (or results) for which exponential functions are performed.

Exponential growth

When a quantity is constantly and proportionately, forming an upward curve in the graph, it can be said that exponential growth is being attained. Descriptively, it can be said that when a specific quantity is increasing throughout time constantly it is exponential growth. There are multiple growth formulas for exponential growth, but mainly:

f(x) = a(1+r)x  is used. Where a is an initial value, r is a rate of growth and x is time.

Economic decay

When a quantity is proportionately forming a downward curve in its graphical representation then it can be termed as exponential decay. Descriptively, if a specific quantity is decreasing proportionately from its current position throughout time, it can be termed as exponential decay. The formula for exponential decay is:

f(x) = a(1-r)x. Where a is an initial value, r is a rate of growth and x is time.

Derivatives

The derivative of exponential function is product of exponential function ax and natural logarithm of a that is d(ax)/dx = (ax)’ = ax lna, where f(x) = ax, a > 0. This derivative can be found from principles of differentiation by definition of limit. Graph of derivative of exponential function changes its place downward and upward when a>1 or a<1.Derivative of exponential function formula:

•     f(x) = ax, f'(x) = ax ln a ; or d(ax)/dx = ax ln a.
•     f(x) = ex, f'(x) = ex ; or d(ex)/dx = ex.

Rules and properties of exponential functions:

1.)  Law of Zero Exponent: a0 = 1

Any number that reaches zero power is equal to one.

2.)  Law of Product: ap × aq = ap+q

Powers should be multiplied with the same base, add the exponents and keep the common base.

3.)  Law of Quotient: ap/aq = ap-q

Powers should be divided with common base, exponents to get subtracted and keep the common base.

4.)  Law of Power of a Power: (ap)q = apq

Power raised to a power, keep the base and multiply the exponents.

5.)  Law of Power of a Product: (ab)p = apbp

The product raised to a power, raises each factor to the power.

6.)  Law of Power of a Quotient: (a/b)p = ap/bp

Quotient should be raised to a power, the numerator and the denominator should be raised to the power.

7.)  Law of Negative Exponent: a-p = 1/ap

Negative exponents go in reciprocation, with the exponent of the reciprocal and then becoming positive.

Equality property of Exponential functions

Exponential functions have a special property called equality property. In this property, if two exponential functions with the same bases are equal, then their exponents are also common. 

i.e.

Examples

1.) Let’s take a = 2

f(x) = ax

f(x) = 2x

f(x) = (1/ 2)x = 2-x

f(x) = 2x+3

f(x) = 0.5x

2.)Let’s take a = 3

f(x) = ax

f(x) = 3x

f(x) = 1/ 3x = 3-x

f(x) = 5x+3

f(x) = 0.6x

Uses of exponential functions

Exponential functions are very useful to prepare real-world population models, carbon date artifacts, compute investments, predict the time of death of coroners, and calculate compound interest-related problems and population growth, etc. exponential growth and decay can be used to determine all problems. Graphical representation of exponential functions can be easily understood by anyone without doing any calculation of the growth or decay.

Conclusion:

Above from here, we came by exponential functions. We learned its definition, formulae, derivatives, properties, rules, uses and multiple examples also. It is highly important to have the exponential function’s box checked to score well for exams, Exponential functions are not somewhat easy topics to understand. This brief note on exponential function can help you to learn and grip numerical functions more easily.