✅ Exponential Equation Formula ⭐️⭐️⭐️⭐️⭐

Exponential Functions

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.

What is Exponential Function?

An exponential function is a Mathematical function in form f (x) = a^x, 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 Formula

An exponential function is defined by the formula f(x) = a^x, 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) = a^x

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 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 Graph

The 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 = f_n(x) increases for increasing values of (n).

From the above, it can be seen that the nature of polynomial functions is dependent on its degree. Higher the degree of any polynomial function, then higher is its growth. A function which grows faster than a polynomial function is y = f(x) = a^x, where a>1. Thus, for any of the positive integers n the function f (x) is said to grow faster than that of f_n(x).

Thus, the exponential function having base greater than 1, i.e., a > 1 is defined as y = f(x) = a^x. 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 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 base the function, we get y = e^x, 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 a^x = b. Thus, log_a b = x if a^x = b. This function is known as logarithmic function.

For base a = 10, this function is known as common logarithm and for the base a = e, it is known as natural logarithm denoted by ln x. Following are some of the important observations regarding logarithmic functions which have a base a>1.

• The domain of log function consists of positive real numbers only, as we cannot interpret the meaning of log functions for negative values.
• For the log function, though the domain is only the set of positive real numbers, the range is set of all real values, i.e. R
• When we plot the graph of log functions and move from left to right, the functions show increasing behaviour.
• The graph of log function never cuts x-axis or y-axis, though it seems to tend towards them.

• Log_ap = α, log_bp = β and log_ba = µ, then a^α = p, b^β = p and b^µ = a
• Log_bpq = Log_bp + Log_bq
• Log_bp^y = ylog_bp
• Log_b (p/q) = log_bp – log_bq

Exponential Function Derivative

Let us now focus on the derivative of exponential functions.

The derivative of e^x with respect to x is e^x, i.e. d(e^x)/dx = e^x

It is noted that the exponential function f(x) =e^x  has a special property. It means that the derivative of the function is the function itself.

(i.e) f ‘(x) = e^x = f(x)

Exponential Series

The exponential series are given below.

Exponential Function Properties

The exponential graph of a function represents the exponential function properties.

Let us consider the exponential function, y=2^x

The graph of function y=2^x is shown below. First, the property of the exponential function graph when the base is greater than 1.

Exponential Function Graph for y=2^x

The graph passes through the point (0,1).

• The domain is all real numbers
• The range is y>0
• The graph is increasing
• The graph is asymptotic to the x-axis as x approaches negative infinity
• The graph increases without bound as x approaches positive infinity
• The graph is continuous
• The graph is smooth

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.

• The line passes through the point (0,1)
• The domain includes all real numbers
• The range is of y>0
• It forms a decreasing graph
• The line in the graph above is asymptotic to the x-axis as x approaches positive infinity
• The line increases without bound as x approaches negative infinity
• It is a continuous graph
• It forms a smooth graph

Exponential Function Rules

Some important exponential rules are given below:

If a>0, and  b>0, the following hold true for all the real numbers x and y:

• a^x a^y = a^x+y
• a^x/a^y = a^x-y
• (a^x)^y = a^xy
• a^xb^x=(ab)^x
• (a/b)^x= a^x/b^x
• a⁰=1
• a^-x= 1/ a^x

Exponential Functions Examples

The examples of exponential functions are:

• f(x) = 2^x
• f(x) = 1/ 2^x = 2^-x
• f(x) = 2^x+3
• f(x) = 0.5^x

Solved problem

Question:

Simplify the exponential equation 2^x-2^x+1

Solution:

Given exponential equation: 2^x-2^x+1

By using the property: a^x a^y = a^x+y

Hence, 2^x+1 can be written as 2^x. 2

 Thus the given equation is written as:

2^x-2^x+1 =2^x-2^x. 2

Now, factor out the term 2^x

2^x-2^x+1 =2^x-2^x. 2 = 2^x(1-2)

2^x-2^x+1 = 2^x(-1)

2^x-2^x+1 = – 2^x

Therefore, the simplification of the given expoential equation  2^x-2^x+1 is  – 2^x.

Exponential Function

The exponential function is 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 an exponential function.

Let us learn more about exponential function along with its definition, equation, graphs, exponential growth, exponential decay, etc.

What is Exponential Function?

Exponential function, as its name suggests, involves 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 Definition

In mathematics, an exponential function is a function of form f (x) = a^x, 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.

Exponential Function Examples

Here are some examples of exponential function.

• f(x) = 2^x
• f(x) = (1/2)^x
• f(x) = 3e^2x
• f(x) = 4 (3)^-0.5x

Exponential Function Formula

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 exponential function 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).

The exponential function arises whenever 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 half-life, 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: f(x) = abx, where b > 1. f(x) = a (1 + r)x P = P${}_0$ ek t Here, b = 1 + r ≈ ek.	In exponential decay, the function can be of the form: f(x) = abx, where 0 < b < 1. f(x) = a (1 – r)x P = P${}_0$ e-k t Here, b = 1 – r ≈ e– k.

In the above formulas,

• a (or) P00 = 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).

Exponential Series

The real exponential function can be commonly defined by the following power series,

Exponential Function Rules

The rules of exponential function are as same as that of rules of exponents. Here are some rules of exponents.

• Law of Zero Exponent: a⁰ = 1
• Law of Product: a^m × aⁿ = a^m+n
• Law of Quotient: a^m/aⁿ = a^m-n
• Law of Power of a Power: (a^m)ⁿ = a^mn
• Law of Power of a Product: (ab)^m = a^mb^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 ⇔ logbb a = x

Equality Property of Exponential Function

According to the equality property of exponential function, if two exponential functions of the same bases are the same, then their exponents are also the same. i.e.,

b^x1 = b^x2 ⇔ x1 = x2

Exponential Function Graph

We can understand the process of graphing exponential function 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 Function

We know that the domain of a function y = f(x) is the set of all x-values (inputs) where it can be computed and the range is the set of all y-values (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 an 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 Function Derivative

Here are the formulas from differentiation that are used to find the derivative of exponential function.

• d/dx (e^x) = e^x
• d/dx (a^x) = a^x · ln a.

Integration of Exponential Function

Here are the formulas from integration that are used to find the integral of exponential function.

• ∫ e^x dx = e^x + C
• ∫ a^x dx = a^x / (ln a) + C

Examples on Exponential Function

What are the Rules of Exponential Function?

How to Graph Exponential Function?

What is the Domain of Exponential Function?

What is the Range of Exponential Function?

What is the Integral of Exponential Function?

Introduction to Exponential Functions

Learning Objective(s)

·         Graph exponential equations and functions.

·         Solve applied problems using exponential problems and their graphs.

Introduction

In addition to linear, quadratic, rational, and radical functions, there are exponential functions.Exponential functions have the form f(x) = b^x, where b > 0 and b ≠ 1. Just as in any exponential expression, b is called the base and x is called the exponent.

An example of an exponential function is the growth of bacteria. Some bacteria double every hour. If you start with 1 bacterium and it doubles every hour, you will have 2^x bacteria after x hours. This can be written as f(x) = 2^x.

Before you start,   f(0) = 2⁰ = 1

After 1 hour           f(1) = 2¹ = 2

In 2 hours              f(2) = 2² = 4

In 3 hours              f(3) = 2³ = 8

and so on.

With the definition f(x) = b^x and the restrictions that b > 0 and that b ≠ 1, the domain of an exponential function is the set of all real numbers. The range is the set of all positive real numbers. The following graph shows f(x) = 2^x.

Exponential Growth

As you can see above, this exponential function has a graph that gets very close to the x-axis as the graph extends to the left (as x becomes more negative), but never really touches the x-axis. Knowing the general shape of the graphs of exponential functions is helpful for graphing specific exponential equations or functions.

Making a table of values is also helpful, because you can use the table to place the curve of the graph more accurately. One thing to remember is that if a base has a negative exponent, then take the reciprocal of the base to make the exponent positive. For example,

Example

Look at the table of values. Think about what happens as the x values increase—so do the function values (f(x) or y)!

Now that you have a table of values, you can use these values to help you draw both the shape and location of the function. Connect the points as best you can to make a smooth curve (not a series of straight lines). This shows that all of the points on the curve are part of this function.

This is an example of exponential growth. As x increases, f(x) “grows” more quickly. Let’s try another one.

Let’s compare the three graphs you’ve seen. The functions f(x) = 2^x, f(x) = 3^x, and

f(x) = 4^x are all graphed below.

Notice that a larger base makes the graph steeper. A larger base also makes the graph closer to the y-axis for x > 0 and closer to the x-axis for x < 0. All the graphs go through (0, 1)!

Exponential Decay

Remember that for exponential functions, b > 0, but b ≠ 1. In the examples above, b > 1. What happens when b is between 0 and 1, 0 < b < 1?

Notice that the shape is similar to the shape when b > 1, but this time the graph gets closer to the x-axis when x > 0, rather than when x < 0. This is exponential decay. Instead of the function values “growing” as x values increase, as they did before, the function values “decay” or decrease as x values increase. They get closer and closer to 0.

Incorrect. This graph is increasing, because the f(x) or y values increase as the x values increase. (Compare the values for x = 1 and x = 2.) This graph shows exponential growth, with a base greater than 1. The correct answer is Graph D.

Incorrect. This graph is decreasing, but all the function values are negative. The range for an exponential function is always positive values. The correct answer is Graph D.

Incorrect. This graph is increasing, but all the function values are negative. The correct graph must be decreasing with positive function values. The correct answer is Graph D.

Correct. All the function values are positive, and the graph is decreasing (showing exponential decay).

Applying Exponential Functions

Exponential functions can be used in many contexts, such as compound interest (money), population growth, and radioactive decay. In most of these, however, the function is not exactly of the form f(x) = b^x. Often, this is adjusted by adding or multiplying constants.

For example, the compound interest formula is

, where P is the principal (the initial investment that is gathering interest) and A is the amount of money you would have, with interest, at the end of t years, using an annual interest rate of r (expressed as a decimal) and m compounding periods per year. In this case the base is the value represented by the expression

Radioactive decay is an example of exponential decay. Radioactive elements have a half-life. This is the amount of time it takes for half of a mass of the element to decay into another substance. For example, uranium-238 is a slowly decaying radioactive element with a half-life of about 4.47 billion years. That means it will take that long for 100 grams of uranium-238 to turn into 50 grams of uranium-238 (the other 50 grams will have turned into another element). That’s a long time! On the other extreme, radon-220 has a half-life of about 56 seconds. What does this mean? 100 grams of radon-220 will turn into 50 grams of radon-220 and 50 grams of something else in less than a minute!

Since the amount is halved each half-life, an exponential function can be used to describe the amount remaining over time. The formula

gives the remaining amount R from an initial amount A, where h is the half-life of the element and t is the amount of time passed (using the same time unit as the half-life).

Billy’s mother put $100 into a bank account for him when he was born. The account gained interest at a rate of 3% per year, compounded monthly. Assuming no more money was deposited and none was withdrawn, how much money will be in the account when Billy turns 18?

A) $170.24

B) $171.49

C) $8561.76

D) $20,718.34

Answer

A) $170.24

Incorrect. This is the amount when the annual rate is compounded annually (m = 1). In this case we are compounding monthly, m = 12. The correct answer is $171.49.

B) $171.49

Correct. Using a rate r of 0.03 and 12 compounding periods per year (m = 12), the formula gives $171.49.

C) $8561.76

Incorrect. You may have used r = 3 rather than r = 0.03 for the annual rate, then misread the result from your calculator. The correct answer is $171.49.

D) $20,718.34

Incorrect. You may have used r = 0.3 rather than r = 0.03 for the annual rate. The correct answer is $171.49.

Summary

Exponential functions of the form f(x) = b^x appear in different contexts, including finance and radioactive decay. The base b must be a positive number and cannot be 1. The graphs of these functions are curves that increase (from left to right) if b > 1, showing exponential growth, and decrease if 0 < b < 1, showing exponential decay.

Example

What Are Exponential Functions?

Before we get into dealing with exponential functions and graphing exponential functions, let’s first take a look at the general formula and theory behind exponential functions.

Below is one of the most general forms of an exponential graph:

formula for exponential functions:

y=ab^d(x−c)+k

The above formula is a little more complicated than previous functions you’ve likely worked with, so let’s define all of the variables.

y – the value on the y-axis

a – the vertical stretch or compression factor

b – the base value

x – the value on the x-axis

c – the horizontal translation factor

d – the horizontal stretch or compression factor

k – the vertical translation factor

In this lesson, we’ll only be going over very basic exponential functions, so you don’t need to worry about some of the above variables. But, so you have access to all of the information you need about exponential functions and how to graph exponential functions, let’s outline what changing each of these variables does to the graph of an exponential equation.

1) Variable “a”

By making this transformation, we have shifted the entire graph to the right two units. If “c” was equal to -2, we would have shifted the entire graph to the left two units.

Variable “d”

How To Find Exponential Functions

Finding the equation of exponential functions is often a multi-step process, and every problem is different based upon the information and type of graph we are given. Given the graph of exponential functions, we need to be able to take some information from the graph itself, and then solve for the stuff we are unable to take directly from the graph. Below is a list of all of the variables we may have to look for, and how to usually find them:

a – solve for it using algebra, or it will be given

b – solve for it using algebra, or it will be given

c – let x = 0 and imagine “c” is not there, the value of y will equal the y-intercept; now count how many units the y value for the y-intercept is from the y-axis, and this will equal “c”

d – solve for it using algebra

k – equal to the value of the horizontal asymptote

Of course, these are just the general steps you need to take in order to find the exponential function equation. The best way to learn how to do this is to try some practice problems!

Exponential Functions Examples:

Now let’s try a couple examples in order to put all of the theory we’ve covered into practice. With practice, you’ll be able to find exponential functions with ease!

Example 1:

Determine the exponential function in the form y=ab^x of the given graph.

In order to solve this problem, we’re going to need to find the variables “a” and “b”. As well, we’re going to have to solve both of these algebraically, as we can’t determine them from the exponential function graph itself.

Step 1: Solve for “a”

To solve for “a”, we must pick a point on the graph where we can eliminate bx because we don’t yet know “b”, and therefore we should pick the y-intercept (0,3). Since b0 equals 1, we can find that a=3. As a shortcut, since we don’t’ have a value for k, a is just equal to the y-intercept of this equation.

Step 2: Solve for “b”

Now that we have “a”, all we have to do is sub in 3 for “a”, pick another point, and solve for b. Let’s pick the point (1,6). With all this information, we can find that b=2.

Step 3: Write the Final Equation

Now we that we have found all of the necessary variables, all that’s left is to write out our final equation in the form y=ab^x. Our final answer is y=(3)2^x

Example 2:

Determine the exponential function in the form y=a2^dx+k of the given graph.

In order to solve this problem, we’re going to need to find the variables “a”, “d” and “k”. Remember, we can find “k” from the graph, as it is the horizontal asymptote. For “a” and “d”, however, we’re going to have to solve for these algebraically, as we can’t determine them from the exponential function graph itself.

Step 1: Find “k” from the Graph

To find “k”, all we need to do is find the horizontal asymptote, which is clearly y=6. Therefore, k=6.

Step 2: Solve for “a”

To solve for “a”, just like the last example, we must pick a point on the graph where we can eliminate 2dx because we don’t yet know “d”, and therefore we should pick the y-intercept (0,3). Since 20 equals 1, subbing (0, 3) into y=a2^dx+6 gives us that a=-3.

Step 3: Solve for “b”

Now that we have “a” and “k”, all we have to do is pick another point and solve for b. Let’s pick the point (0.25, 0). With all this information, we can find that d=4.

Step 4: Write the Final Equation

Now we that we have found all of the necessary variables, all that’s left is to write out our final equation in the form y=ab^dx+k. Our final answer is y=(−3)2^4x+6

And that’s it for exponential functions! Again, these functions are a little more complex than equations for lines or parabolas, so be sure to do lots of practice problems to get a hang of the new variables and techniques. With more practice, soon exponential equations and the graphs of exponential functions will be no trouble at all!

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