An exponential function is a mathematical function, which is used in many real-world situations. It is mainly used to find the exponential decay or exponential growth or to compute investments, model populations and so on. In this article, you will learn about exponential function formulas, rules, properties, graphs, derivatives, exponential series and examples. Show
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What is Exponential Function?An exponential function is a Mathematical function in the form f (x) = ax, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0. The most commonly used exponential function base is the transcendental number e, which is approximately
equal to 2.71828. Exponential Function FormulaAn exponential function is defined by the formula f(x) = ax, where the input variable x occurs as an exponent. The exponential curve depends on the exponential function and it depends on the value of the x. The exponential function is an important mathematical function which is of the form f(x) = ax Where a>0 and a is not equal to 1. x is any real number. If the variable is negative, the function is undefined for -1 < x < 1. Here, “x” is a variable “a” is a constant, which is the base of the function. An exponential curve grows, or decay depends on the exponential function. Any quantity that grows or decays by a fixed per cent at regular intervals should possess either exponential growth or exponential decay. Exponential Growth In Exponential Growth, the quantity increases very slowly at first, and then rapidly. The rate of change increases over time. The rate of growth becomes faster as time passes. The rapid growth is meant to be an “exponential increase”. The formula to define the exponential growth is: y = a ( 1+ r )x Where r is the growth percentage. Exponential Decay In Exponential Decay, the quantity decreases very rapidly at first, and then slowly. The rate of change decreases over time. The rate of change becomes slower as time passes. The rapid growth meant to be an “exponential decrease”. The formula to define the exponential growth is: y = a ( 1- r )x Where r is the decay percentage.
Exponential Function GraphThe following figure represents the graph of exponents of x. It can be seen that as the exponent increases, the curves get steeper and the rate of growth increases respectively. Thus, for x > 1, the value of y = fn(x) increases for increasing values of (n). From the above, it can be seen that the nature of polynomial functions is dependent on their degree. The higher the degree of any polynomial function, the higher its growth. A function which grows faster than a polynomial function is y = f(x) = ax, where a>1. Thus, for any of the positive integers n the function f (x) is said to grow faster than that of fn(x). Thus, the exponential function having base greater than 1, i.e., a > 1 is defined as y = f(x) = ax. The domain of exponential function will be the set of entire real numbers R and the range are said to be the set of all the positive real numbers. It must be noted that the exponential function is increasing and the point (0, 1) always lies on the graph of an exponential function. Also, it is very close to zero if the value of x is mostly negative. Exponential function having base 10 is known as a common exponential function. Consider the following series: The value of this series lies between 2 & 3. It is represented by e. Keeping e as the base of the function, we get y = ex, which is a very important function in mathematics known as a natural exponential function. For a > 1, the logarithm of b to base a is x if ax = b. Thus, loga b = x if ax = b. This function is known as logarithmic function. For base a = 10, this function is known as a common logarithm and for the base a = e, it is known as a natural logarithm denoted by ln x. Following are some of the important observations regarding logarithmic functions which have a base a>1.
Exponential Function DerivativeLet us now focus on the derivative of exponential functions. The derivative of ex with respect to x is ex, i.e. d(ex)/dx = ex It is noted that the exponential function f(x) =ex has a special property. It means that the derivative of the function is the function itself. (i.e) f ‘(x) = ex = f(x) Exponential SeriesThe exponential series are given below. Exponential Function PropertiesThe exponential graph of a function represents the exponential function properties. Let us consider the exponential function, y = 2x The graph of function y = 2x is shown below. Exponential Function Graph for y = 2x The properties of the exponential function graph when the base is greater than 1 are given below.
Exponential Function Graph y=2-x The graph of function y=2-x is shown above. The properties of the exponential function and its graph when the base is between 0 and 1 are given.
Exponential Function RulesSome important exponential rules are given below: If a>0, and b>0, the following hold true for all the real numbers x and y:
Exponential Functions ExamplesThe examples of exponential functions are:
Solved ProblemsQuestion 1: Simplify the exponential function 2x – 2x+1 Solution: Given exponential function: 2x – 2x+1 By using the property: ax ay = ax+y Hence, 2x+1 can be written as 2x. 2 Thus the given function is written as: 2x-2x+1 = 2x-2x. 2 Now, factor out the term 2x 2x-2x+1 = 2x-2x. 2 = 2x(1-2) 2x-2x+1 = 2x(-1) 2x-2x+1 = – 2x Therefore, the simplification of the given exponential function 2x – 2x+1 is – 2x. Question 2: Solve the exponential equation: (¼)x = 64 Solution: Given exponential equation is: (¼)x = 64 Using the exponential rule (a/b)x = ax/bx, we get; 1x/4x = 43 1/4x = 43 [since 1x = 1] (1)(4-x) = 43 4-x = 43 Here, bases are equal. So, by equating the powersm we have; x = -3 Practice Questions
What is the B value in an exponential function?Exponential functions have the form f(x) = bx, where b > 0 and b ≠ 1. Just as in any exponential expression, b is called the base and x is called the exponent.
What does the A value mean in exponential equation?Exponential functions are based on relationships involving a constant multiplier. You can write. an exponential function in general form. In this form, a represents an initial value or amount, and b, the constant multiplier, is a growth factor or factor of decay.
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