Exponential Functions
Exponential functions are a type of mathematical function which are helpful in finding the growth or decay of population, money, price, etc that are growing or decay exponentially. Jonathan was reading a news article on the latest research made on bacterial growth. He read that an experiment was conducted with one bacterium. After the first hour, the bacterium doubled itself and was two in number. After the second hour, the number was four. At every hour the number of bacteria was increasing. He was thinking what would be the number of bacteria after 100 hours if this pattern continues. When he asked his teacher about the same the answer he got was the concept of exponential functions.
Let us learn more about exponential functions along with their definition, equation, graphs, exponential growth, exponential decay, etc.
What Are Exponential Functions?
Exponential functions, as its name suggests, involve exponents. But note that, an exponential function has a constant as its base and a variable as its exponent but not the other way round (if a function has a variable as the base and a constant as the exponent then it is a power function but not an exponential function). An exponential function can be in one of the following forms.
Exponential Function Formulas
A basic exponential function, from its definition, is of the form f(x) = b^{x}, where 'b' is a constant and 'x' is a variable. One of the popular exponential functions is f(x) = e^{x}, where 'e' is "Euler's number" and e = 2.718....If we extend the possibilities of different exponential functions, an exponential function may involve a constant as a multiple of the variable in its power. i.e., an exponential function can also be of the form f(x) = e^{kx}. Further, it can also be of the form f(x) = p e^{kx}, where 'p' is a constant. Thus, an exponential function can be in one of the following forms.
 f(x) = b^{x}
 f(x) = ab^{x}
 f(x) = ab^{cx}
 f(x) = e^{x}
 f(x) = e^{kx}
 f(x) = p e^{kx}
Here, apart from 'x' all other letters are constants, 'x' is a variable, and f(x) is an exponential function in terms of x. Also, note that the base in each of the exponential functions must be a positive number. i.e., in the above functions, b > 0 and e > 0. Also, b should not be equal to 1 (if b = 1, then the function f(x) = b^{x} becomes f(x) = 1 and in this case, the function is linear but NOT exponential).
Examples of Exponential Functions
Here are some examples of exponential functions.
 f(x) = 2^{x}
 f(x) = (1/2)^{x}
 f(x) = 3e^{2x}
 f(x) = 4 (3)^{0.5x}
Rules of Exponential Functions
The rules of exponential functions are as same as that of rules of exponents. Here are some rules of exponents.
 Law of Zero Exponent: a^{0} = 1
 Law of Product: a^{m} × a^{n} = a^{m+n}
 Law of Quotient: a^{m}/a^{n} = a^{mn}
 Law of Power of a Power: (a^{m})^{n} = a^{mn}
 Law of Power of a Product: (ab)^{m} = a^{m}b^{m}
 Law of Power of a Quotient: (a/b)^{m} = a^{m}/b^{m}
 Law of Negative Exponent: a^{m} = 1/a^{m}
Apart from these, we sometimes need to use the conversion formula of logarithmic form to exponential form which is:
 b^{x }= a ⇔ log\(_b\) a = x
Equality Property of Exponential Functions
According to the equality property of exponential functions, if two exponential functions of the same bases are the same, then their exponents are also the same. i.e.,
b^{\(x_1\)} = b^{\(x_2\) }⇔ \(x_1\) = \(x_2\)
Graphing Exponential Functions
We can understand the process of graphing exponential functions by taking some examples. Let us graph two functions f(x) = 2^{x} and g(x) = (1/2)^{x}. To graph each of these functions, we will construct a table of values with some random values of x, plot the points on the graph, connect them by a curve, and extend the curve on both ends.
Here is the table of values that are used to graph the exponential function f(x) = 2^{x}.
Here is the table of values that are used to graph the exponential function g(x) = (1/2)^{x}.
Note: From the above two graphs, we can see that f(x) = 2^{x} is increasing whereas g(x) = (1/2)^{x} is decreasing. Thus, the graph of exponential function f(x) = b^{x}
 increases when b > 1
 decreases when 0 < b < 1
Domain and Range of Exponential Functions
We know that the domain of a function y = f(x) is the set of all xvalues (inputs) where it can be computed and the range is the set of all yvalues (outputs) of the function. From the graphs of f(x) = 2^{x} and g(x) = (1/2)^{x} in the previous section, we can see that an exponential function can be computed at all values of x. Thus, the domain of any exponential function is the set of all real numbers (or) (∞, ∞). The range of an exponential function can be determined by the horizontal asymptote of the graph, say, y = d, and by seeing whether the graph is above y = d or below y = d. Thus, for an exponential function f(x) = ab^{x},
 Domain is the set of all real numbers (or) (∞, ∞).
 Range is f(x) > d if a > 0 and f(x) < d if a < 0.
To understand this, you can see the example below.
Exponential Growth and Exponential Decay Formulas
As the name suggests, a quantity's value increases in exponential growth and decreases in exponential decay. We can see more differences between exponential growth and decay along with their formulas in the following table.
Exponential Growth  Exponential Decay 

In exponential growth, a quantity slowly increases in the beginning and then it increases rapidly.  In exponential decay, a quantity decreases very rapidly in the beginning, and then it decreases slowly. 
The exponential growth formulas are used to model population growth, to model compound interest, to find doubling time, etc  The exponential decay is helpful to model population decay, to find halflife, etc. 
The graph of the function in exponential growth is increasing.  The graph of the function in exponential growth is decreasing. 
In exponential growth, the function can be of the form:
Here, b = 1 + r ≈ e^{k}. 
In exponential decay, the function can be of the form:
Here, b = 1  r ≈ e^{ k}. 
In the above formulas,
 a (or) P\(_0\) = Initial value
 r = Rate of growth
 k = constant of proportionality
 x (or) t = time (time can be in years, days, (or) months. Whatever we are using should be consistent throughout the problem).
Derivatives and Integrals of Exponential Functions
Here are the formulas from differentiation that are used to find the derivatives of exponential functions.
 d/dx (e^{x}) = e^{x}
 d/dx (a^{x}) = a^{x} · ln a.
Here are the formulas from integration that are used to find the integrals of exponential functions.
 ∫ e^{x} dx = e^{x} + C
 ∫ a^{x} dx = a^{x} / (ln a) + C
Examples on Exponential Functions

Example 1: In 2010, there were 100,000 citizens in a town. If the population increases by 8% every year, then how many citizens will there be in 10 years? Round your answer to the nearest integer.
Solution:
The initial population is, a = 100,000.
The rate of growth is, r = 8% = 0.08.
The time is, x = 10 years.
Using the exponential growth formula,
f(x) = a (1 + r)^{x}
f(x) = 100000(1 + 0.08)^{10}
f(x) ≈ 215,892 (rounded to the nearest integer)
Answer: Therefore, the number of citizens in 10 years will be 215,892.

Example 2: The halflife of carbon14 is 5,730 years. If there were 1000 grams of carbon initially, then what is the amount of carbon left after 2000 years? Round your answer to the nearest integer.
Solution:
Using the given data, we can say that carbon14 is decaying and hence we use the formula of exponential decay.
P = P\(_0\) e^{ k t} ... (1),
Here, P\(_0\) = initial amount of carbon = 1000 grams.
It is given that the halflife of carbon14 is 5,730 years. It means
P = P\(_0\) / 2 = 1000 / 2 = 500 grams.
Substitute all these values in (1),
500 = 1000 e^{ k (5730)}
Dividing both sides by 1000,
0.5 = e^{ k (5730)}
Taking "ln" on both sides,
ln 0.5 = 5730k
Dividing both sides by 5730,
k = ln 0.5 / (5730) ≈ 0.00012097
We have to find the amount of carbon that is left after 2000 years. Substitute t = 2000 in (1),
P = 1000 e^{ (0.00012097) (2000)} ≈ 785 grams.
Answer: The amount of carbon left after 1000 years = 785 grams.
FAQs on Exponential Functions
What Are Exponential Functions?
An exponential function is a type of function in math that involves exponents. A basic exponential function is of the form f(x) = b^{x}, where b > 0 and b ≠ 1.
What Ars the Formulas of Exponential Function?
The formulas of an exponential function have exponents in them. An exponential equation can be in one of the following forms.
 f(x) = ab^{cx}
 f(x) = p e^{kx}
What Are the Rules of Exponential Functions?
Since the exponential functions involve exponents, the rules of exponential functions are as same as the rules of exponents. They are:
 a^{m} × a^{n} = a^{m+n}
 a^{m}/a^{n} = a^{mn}
 a^{0} = 1
 a^{m} = 1/a^{m}
 (a^{m})^{n} = a^{mn}
 (ab)^{m} = a^{m}b^{m}
 (a/b)^{m} = a^{m}/b^{m}
How To Graph Exponential Functions?
To graph an exponential function y = f(x), create a table of values by taking some random numbers for x (usually we take 2, 1, 0, 1, and 2), substitute each of them in the function to find the corresponding y values. Then plot the points from the table and join them by a curve. Finally, extend the curve on both ends.
What Is the Domain of Exponential Functions?
An exponential function f(x) = ab^{x} is defined for all values of x and hence its domain is the set of all real numbers, which in interval notation can be written as (∞, ∞).
What Is the Range of Exponential Functions?
The range of an exponential function depends upon its horizontal asymptote and also whether the curve lies above or below the horizontal asymptote. i.e., for an exponential function f(x) = ab^{x}, the range is
 f(x) > d if a > 0 and
 f(x) < d if a < 0,
where y = d is the horizontal asymptote of the graph of the function.
What Is the Equality Property of Exponential Functions?
The equality property of exponential functions says if two values (outputs) of an exponential function are equal, then the corresponding inputs are also equal. i.e., b^{\(x_1\)} = b^{\(x_2\) }⇔ \(x_1\) = \(x_2\).
What Are the Derivatives of Exponential Functions?
An exponential function may be of the form e^{x} or a^{x}. The formulas to find the derivatives of these functions are as follows:
 d/dx (e^{x}) = e^{x}
 d/dx (a^{x}) = a^{x} · ln a.
What Are the Integrals of Exponential Functions?
An exponential function may be of the form e^{x} or a^{x}. The formulas to find the integrals of these functions are as follows:
 ∫ e^{x} dx = e^{x} + C
 ∫ a^{x} dx = a^{x} / (ln a) + C