Exponential functions come up often in algebra and calculus. It is one thing to make a table or graph given the formula for an exponential function, but working backwards is another thing.

So, how do you find the formula of an exponential function? **We can find the formula of an exponential function by using two points on the curve, substituting them into the formula y = ab ^{x}, and solving the system of two equations in two unknowns. Given a graph or a table of values, we just need to choose two points and use the same method described above.**

Of course, we cannot find the formula of an exponential function from one point, since one point (one equation) does not give us enough information to solve for two parameters (the coefficient a and the base b).

In this article, we’ll take a look at the ways you can find the formula of an exponential function. We’ll also dive into a few examples to make the concepts clear.

Let’s get started.

## How To Find The Formula Of An Exponential Function

To find the formula of an exponential function, all we need is two different points on the curve. Once we have those two points, we can substitute each pair into the general formula for an exponential function:

**f(x) = ab**^{x}(or y = ab^{x})

After we substitute both points, we get two equations in two unknowns (the parameters a and b). We can then solve the system of equations.

If you are given a table of values for an exponential function, you can simply choose two points and use the same method outlined above.

If you are given the graph of an exponential function, you can simply choose two points on the curve and use the same method outlined above.

It all comes down to finding two points on the curve and using them to write two equations, then solving the system.

Let’s take a closer look at each method and some examples.

### How To Find The Formula Of An Exponential Function With Two Points

Remember, there are three basic steps to find the formula of an exponential function with two points:

**1. Plug in the first point into the formula y = ab**^{x}to get your first equation.**2. Plug in the second point into the formula y = ab**^{x}to get your second equation.**3. Solve the system of two equations that you got from steps 1 & 2. This will give you a and b.**

Here are some examples to show how it works.

#### Example 1: How To Find The Formula Of An Exponential Function With Two Points

Let’s say that you are given the points (0, 1) and (2, 9), which lie on the graph of an exponential function.

We know that the general formula for an exponential function is given by:

**f(x) = ab**^{x}(or y = ab^{x})

Using the first point (0, 1), we substitute x = 0 and y = 1 to get:

**y = ab**^{x}**1 = ab**^{0}**1 = a*1****1 = a**

In this case, the exponent of 0 on b causes it to cancel out, which gives us a = 1.

Now, using the second point (2, 9), we substitute x = 2 and y = 9 to get:

**y = ab**^{x}**9 = b**^{2}**9 = 1*b**[since a = 1, as we found before]^{2}**3 = b**[take the principal (or positive) square root of 9 to get 3]

So, we have b = 3. Using both a = 1 and b = 3 in the general formula for an exponential function, we get:

**y = ab**^{x}**y = 1*3**^{x}**y = 3**^{x}

So, the exponential function in this case is y = 3^{x} or f(x) = 3^{x}.

You can see the graph of this function below, which includes the two points (0, 1) and (2, 9).

#### Example 2: How To Find The Formula Of An Exponential Function With Two Points

Let’s say that you are given the points (1, 10) and (3, 40), which lie on the graph of an exponential function.

We know that the general formula for an exponential function is given by:

**f(x) = ab**^{x}(or y = ab^{x})

Using the first point (1, 10), we substitute x = 1 and y = 10 to get:

**y = ab**^{x}**10 = ab**^{1}**10 = ab**

Now, using the second point (3, 40), we substitute x = 3 and y = 40 to get:

**y = ab**^{x}**40 = ab**^{3}

So, our system of two equations in two unknowns is:

**10 = ab****40 = ab**^{3}

To solve this system, we will solve for a in the first equation and substitute into the second.

Solving the first equation for a gives us:

**10 = ab****10/b = a**

Substituting into the second equation gives us:

**40 = ab**^{3}**40 = (10/b)b**^{3}**40 = 10b**^{3-1}**40 = 10b**^{2}**40/10 = b**^{2}**4 = b**^{2}**2 = b**[take the principal (or positive) square root of 4 to get 2]

So, we have b = 2. Substituting into our equation for a, we get:

**10/b = a****10/2 = a****5 = a**

Using both a = 5 and b = 2 in the general formula for an exponential function, we get:

**y = ab**^{x}**y = 5*2**^{x}

So, the exponential function in this case is y = 5*2^{x} or f(x) = 5*2^{x}.

You can see the graph of this function below, which includes the two points (1, 10) and (3, 40).

### How To Find The Formula Of An Exponential Function Given A Table

If we are given a table for an exponential function, we can just pick any two points. Then, we can solve for the formula of the exponential function in the same manner as above.

Let’s try an example to see how it works.

#### Example 3: How To Find The Formula Of An Exponential Function Given A Table

Let’s say that you are given the following table of points, which lie on the graph of an exponential function.

x | y |
---|---|

2 | 63 |

3 | 189 |

4 | 567 |

5 | 1707 |

Let’s choose the two points (2, 63) and (4, 567).

We know that the general formula for an exponential function is given by:

**f(x) = ab**^{x}(or y = ab^{x})

Using the first point (2, 63), we substitute x = 2 and y = 63 to get:

**y = ab**^{x}**63 = ab**^{2}

Now, using the second point (4, 567), we substitute x = 4 and y = 567 to get:

**y = ab**^{x}**567 = ab**^{4}

So, our system of two equations in two unknowns is:

**63 = ab**^{2}**567 = ab**^{4}

To solve this system, we will solve for a in the first equation and substitute into the second.

Solving the first equation for a gives us:

**63 = ab**^{2}**63/b**^{2}= a

Substituting into the second equation gives us:

**567 = ab**^{4}**567 = (63/b**^{2})b^{4}**567 = 63b**^{4-2}**567 = 63b**^{2}**567/63 = b**^{2}**9 = b**^{2}**3 = b**[take the principal (or positive) square root of 9 to get 3]

So, we have b = 3. Substituting into our equation for a, we get:

**63/b**^{2}= a**63/3**^{2}= a**63/9 = a****7 = a**

Using both a = 7 and b = 3 in the general formula for an exponential function, we get:

**y = ab**^{x}**y = 7*3**^{x}

So, the exponential function in this case is y = 7*3^{x} or f(x) = 7*3^{x}.

You can see the graph of this function below, which includes the two points (2, 63) and (4, 567).

### How To Find An Exponential Function From A Graph

If we are given the graph for an exponential function, we can just pick any two points on the curve. Then, we can solve for the formula of the exponential function in the same manner as above.

Let’s try an example to see how it works.

#### Example 4: How To Find The Formula Of An Exponential Function Given A Table

Let’s say that you are given the following graph of an exponential function.

We can see that the two points (0, 3) and (1, 15) are on the graph.

We know that the general formula for an exponential function is given by:

**f(x) = ab**^{x}(or y = ab^{x})

Using the first point (0, 3), we substitute x = 0 and y = 3 to get:

**y = ab**^{x}**3 = ab**^{0}**3 = a*1****3 = a**

In this case, the exponent of 0 on b causes it to cancel out, which gives us a = 3.

Now, using the second point (1, 15), we substitute x = 1 and y = 15 to get:

**y = ab**^{x}**15 = 3*b**[since a = 3, as we found before]^{1}**15/3 = b**[since a = 1, as we found before]**5 = b**

So, we have b = 5. Using both a = 3 and b = 5 in the general formula for an exponential function, we get:

**y = ab**^{x}**y = 3*5**^{x}

So, the exponential function in this case is y = 3*5^{x} or f(x) = 3*5^{x}.

### How Do You Find The Base Of An Exponential Function?

To find the base “b” of an exponential function, we still need two points, as before. However, we can use the following formula to find the base b:

**b = (y**_{2}/y_{1})^(1/(x_{2}– x_{1}))

where the two points on the exponential function curve are (x_{1, }y_{1}) and (x_{2, }y_{2}).

Here is where the above formula comes from:

First, we take the general formula for an exponential function:

**y = ab**^{x}

Next, we plug in the first point (x_{1, }y_{1}) to get:

**y = ab**^{x}**y**_{1}= ab^{x1}

Then, we plug in the second point (x_{2, }y_{2}) to get:

**y = ab**^{x}**y**_{2}= ab^{x2}

Now, we divide y_{2} by y_{1} to get:

**y**_{2}/y_{1}= ab^{x2}/ab^{x1}**y**[since the a cancels in the numerator and denominator]_{2}/y_{1}= b^{x2}/b^{x1}_{ }**y**[by the rules of exponents]_{2}/y_{1}= b^{x2-x1}**(y**[take the root with index x_{2}/y_{1})^(1/(x_{2}– x_{1})) = b_{2}– x_{1}to solve for b]

After we find the base b, we can solve for a by using either of the two points given.

Let’s try an example to see how it works.

#### Example 5: How To Find The Base Of An Exponential Function (From Two Points)

Let’s say we have the points (2, 98) and (3, 686) on an exponential function. Then our values are:

**x**_{1}= 2**y**_{1}= 98**x**_{2}= 3**y**_{2}= 686

We use these values and the formula from before to find b:

**b = (y**_{2}/y_{1})^(1/(x_{2}– x_{1}))**b = (686/98)^(1/(3 – 2))****b = (7)^(1/(1))****b = (7)^(1)****b = 7**

Now that we have b = 7, we can use the general formula for an exponential function and the point (2, 98) to find a:

**y = ab**^{x}**98 = a*7**^{2}[b = 7, x = 2, y = 98]**98 = a*49****2 = a**

So a = 2. Using a = 2 and b = 7 in the general formula for an exponential function, we get:

**y = ab**^{x}**y = 2*7**^{x}

So, the exponential function in this case is y = 2*7^{x} or f(x) = 2*7^{x}.

You can see the graph of this function below, which includes the two points (2, 98) and (3, 686).

## Conclusion

Now you know how to find the formula of an exponential function from two points. You also know how to choose two points from a table or graph, and how to use a shortcut formula to find the base “b”.

You can learn more about the domain and range of exponential functions here.

You can learn more about the domain of functions (and how to find it) here.

You can learn about linear vs exponential growth here.

You can learn more about the natural base e ~ 2.718 here.

I hope you found this article helpful. If so, please share it with someone who can use the information.

Don’t forget to subscribe to my YouTube channel & get updates on new math videos!

Subscribe To My YouTube Channel!

~Jonathon

## FAQs

### What are the three methods we can use to solve an exponential equation? ›

Make use of the above property if you are unable to express both sides of the equation in terms of the same base. Step 1: Isolate the exponential and then apply the logarithm to both sides. Step 2: Apply the power rule for logarithms and write the exponent as a factor of the base. Step 3: Solve the resulting equation.

**How do you write an exponential equation with 3 points? ›**

Form you can write it as 5 over 4 times 2 to the power of X. Plus three now the key is noticing that

**How do you find the formula for an exponential function? ›**

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. Where a>0 and a is not equal to 1. x is any real number.

**Are there different methods to solve an exponential equation? ›**

**There are two methods for solving exponential equations**. One method is fairly simple but requires a very special form of the exponential equation. The other will work on more complicated exponential equations but can be a little messy at times.

**How do you solve exponential equations step by step? ›**

Solving Exponential Equations [fbt] (Step-by-Step) - YouTube

**How do you find an exponential function from points? ›**

Ex: Find an Exponential Function Given Two Points - Initial Value Not ...

**How do you find the exponential function from a set of points? ›**

If you have two points, (x_{1}, y_{1}) and (x_{2}, y_{2}), you can define the exponential function that passes through these points by **substituting them in the equation y = ab ^{x} and solving for a and b**. In general, you have to solve this pair of equations: y

_{1}= ab

^{x1}and y

_{2}= ab

^{x2}

^{,}.

**How do you write an equation for an exponential function from a table? ›**

Exponential Equation Given a Table - YouTube

**How do you find the equation of an exponential function from a graph? ›**

How to find equation of exponential Function from Graph - YouTube

**How do you write an equation for an exponential function given two points? ›**

Write an exponential function given two points - YouTube

### How do you solve exponential equations without common bases? ›

In general we can solve exponential equations whose terms do not have like bases in the following way: **Apply the logarithm to both sides of the equation**. If one of the terms in the equation has base 10 , use the common logarithm. If none of the terms in the equation has base 10 , use the natural logarithm.

**How do you solve exponential equations without a calculator? ›**

Master Solving Exponential equations without using a calculator - YouTube

**How do you solve exponential equations with unknown bases? ›**

Rewrite each side in the equation as a power with a common base. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form bS=bT b S = b T . Use the one-to-one property to set the exponents equal to each other. Solve the resulting equation, S = T, for the unknown.

**How do you solve exponential equations word problems? ›**

Word Problems with Exponential Functions - YouTube

**How do you solve exponential equations with fractional bases? ›**

Solve an exponential equation when your base is a fraction - YouTube

**How do you solve equations with 4 powers? ›**

How To Solve a Polynomial Equation Raised to the Fourth Power

**How do you solve an exponential equation using the properties of exponents? ›**

Solving exponential equations using exponent properties - YouTube

**What is exponential function and example? ›**

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.

**WHAT IS A in exponential function? ›**

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. Is it Growth or Decay?

**How do you find an exponential equation from ordered pairs? ›**

Writing an Exponential Function Rule Given a Table of Ordered Pairs

### How do you write an exponential rule? ›

In an exponential function, f(x)=a⋅bx f ( x ) = a ⋅ b x , the growth factor is b .

**How do you find the equation of an exponential graph with one point? ›**

Finding an Exponential Function Given 1 Point - YouTube

**How do you solve exponential equations with different bases? ›**

Solving Exponential Equations With Different Bases Using Logarithms

**How do you solve exponential and logarithmic equations? ›**

Solving Exponential and Logarithmic Equations - YouTube

**How can logarithms be used to solve exponential equations? ›**

Taking logarithms will allow us to take advantage of the log rule that says that powers inside a log can be moved out in front as multipliers. **By taking the log of an exponential, we can then move the variable (being in the exponent that's now inside a log) out in front, as a multiplier on the log**.

**What is the first step in solving exponential equations and inequalities? ›**

**Steps for Solving an Equation involving Exponential Functions**

- Isolate the exponential function.
- If convenient, express both sides with a common base and equate the exponents. Otherwise, take the natural log of both sides of the equation and use the Power Rule.

**How do you solve exponential equations without logarithms? ›**

To solve exponential equations without logarithms, **you need to have equations with comparable exponential expressions on either side of the "equals" sign**. Then you can compare the powers and solve.

**How do you solve exponential equations without common bases? ›**

In general we can solve exponential equations whose terms do not have like bases in the following way: **Apply the logarithm to both sides of the equation**. If one of the terms in the equation has base 10 , use the common logarithm. If none of the terms in the equation has base 10 , use the natural logarithm.

**How do you solve exponents without a calculator? ›**

Master Solving Exponential equations without using a calculator - YouTube

**How do you solve 3 logarithmic equations? ›**

Solving Logarithmic Equations - YouTube

### How do you solve a natural exponential equation? ›

- Step 1: Isolate the natural base exponent. ...
- Step 2: Select the appropriate property to isolate the x-variable. ...
- Step 3: Apply the Property and solve for x. ...
- Step 1: Isolate the natural base exponent. ...
- Step 2: Select the appropriate property to isolate the x-variable. ...
- Step 3: Apply the Property and solve for x.

**How do you solve exponential and logarithmic equations and inequalities? ›**

Exponential and Logarithmic Equations and Inequalities - YouTube

**How do you write in exponential form? ›**

In exponential notation, a number usually is expressed as a coefficient between one and ten times an integral power of ten, the exponent. To express a number in exponential notation, write it in the form: **c × 10n**, where c is a number between 1 and 10 (e.g. 1, 2.5, 6.3, 9.8) and n is an integer (e.g. 1, -3, 6, -2).

**How do you solve exponents with variables? ›**

Introduction to Solving Exponents with Variables (Simplifying Math)