Welcome to the **exponential function calculator**, where we'll show you how to solve exponential functions with the points on its line — or simply help you evaluate your exponential function at a specific $x$x or $y$y value.

In this article, we'll cover the following topics:

- What
**exponential functions**are; - What the
**formula for exponential functions**is; and - How to solve an exponential function from two points.

🙋 Care to learn more about the algebraic concept of **exponents**? Then head on over to our exponent calculator!

## How to use the exponential function calculator?

The exponential function calculator can help you solve your exponential function's parameters or help you pinpoint an exact point on the line. Here's how:

First, decide whether you want to **solve** or **evaluate** the function.

When the exponential function calculator is in

**"solve the function"**mode:Decide the function formula shape (e.g., $b^x$bx or $p\cdot e^{kx}$p⋅ekx).

Give the exponential function calculator some $x, y$x,y points that you know are on that line.

The calculator will solve the unknowns in the equation and report back.

When in

**"evaluate the function"**mode:Enter the exponential function's

**parameters**.Enter the $x$x-value to find its corresponding function output.

💡 **Did you know** that you can enter `e`

into a field to use **Euler's number**, $e=2.71828...$e=2.71828...? Try using it for the base $b$b in "evaluate" mode!

## What is an exponential function?

An **exponential function** is a function that raises some constant to its argument. In many applications, the constant is Euler's number, *e*, but other constants can be used, too, depending on what you're calculating.

🙋 **Euler's number** has many uses — learn more about them at our eee calculator!

## What is the exponential function formula?

There is no one-and-only exponential function formula — instead, a function is "exponential" if the argument ($x$x) is used as an exponent to some constant. The most **basic exponential function** looks like this:

$\footnotesizef(x) = b^x$f(x)=bx

We can make matters much more complicated by allowing the exponent to be scaled and shifted, as well as scaling and shifting the result of the exponential:

$\footnotesizef(x) = a\cdot b^{c\cdot x + p} + q$f(x)=a⋅bc⋅x+p+q

Our exponential function calculator can solve the following exponential function formulas:

- $f(x) = b^x$f(x)=bx
- $f(x) = a \cdot b^x$f(x)=a⋅bx
- $f(x) = e^{kx}$f(x)=ekx
- $f(x) = p \cdot e^{kx}$f(x)=p⋅ekx

Other, more complex formulas (like $f(x) = a \cdot b^{c\cdot x + p} + q$f(x)=a⋅bc⋅x+p+q) cannot be solved directly and often have more than one solution.

🙋 Want to learn more about **probabilities**? The exponential distribution calculator can help!

## How do I find the exponential function from two points?

To **find the exponential function** from two points, **plug the points' coordinate values into the equation** and **solve for the constants**. But be aware — if your function uses too many constants, two points won't be enough!

With two points $(x_1, y_1)$(x1,y1) and $(x_2, y_2)$(x2,y2), you can easily solve for $a$a and $b$b — in other words, **you can solve:**

$\footnotesizef(x) = a\cdot b^x$f(x)=a⋅bx

To do this, we'll use our knowledge of exponents. We know the formula's rough shape is $ab^x$abx, so we can say that:

$\footnotesize\begin{split}y_1 &= a\cdot b^{x_1} \\y_2 &= a\cdot b^{x_2} \\\end{split}$y1y2=a⋅bx1=a⋅bx2

and from there, we can determine that:

$\footnotesize\begin{aligned}\frac{a \cdot b^{x_1}}{a \cdot b^{x_2}} &= \frac{y_1}{y_2} \\[0.9em]b^{(x_1-x_2)} &= y_1/y_2 \\\end{aligned} \\[0.3em]\therefore b = (y_1/y_2)^{(x_1-x_2)^{-1}}$a⋅bx2a⋅bx1b(x1−x2)=y2y1=y1/y2∴b=(y1/y2)(x1−x2)−1

Now that we have $b$b, we can quickly determine $a$a:

$\footnotesizea = y_1/b^{x_1} = y_2/b^{x_2}$a=y1/bx1=y2/bx2

One crucial detail: $x_1$x1 and $x_2$x2 may not be equal. However, $y_1$y1 and $y_2$y2 may be the same, but it will mean that $b=1$b=1. Given two points, the **exponential function calculator** will tell you if you've broken any of these rules.

🔎 **Challenge** — can you figure out why that is?

### What if the base is $e$e?

When we know that **the base** $b$b **is Euler's number**, i.e., that $b = e = 2.71828...$b=e=2.71828..., we can go a little further — using only **two points**, we can solve the following equation:

$\footnotesizef(x) = a \cdot e ^{c \cdot x}$f(x)=a⋅ec⋅x

Here's how:

$\footnotesize\begin{split}y_1 &= a\cdot e^{c\cdot x_1} \\y_2 &= a\cdot e^{c\cdot x_2} \\[0.5em]\therefore \frac{a\cdot e^{c\cdot x_1}}{a\cdot e^{c\cdot x_2}} &= \frac{y_1}{y_2} \\[0.75em]e^{c(x_1 - x_2)} &= y_1 / y_2 \\c(x_1 - x_2) &= \log(y_1 / y_2) \\c &= \log(y_1/y_2) / (x_1-x_2) \\\end{split}$y1y2∴a⋅ec⋅x2a⋅ec⋅x1ec(x1−x2)c(x1−x2)c=a⋅ec⋅x1=a⋅ec⋅x2=y2y1=y1/y2=log(y1/y2)=log(y1/y2)/(x1−x2)

And with $c$c, we can easily say that:

$\footnotesizea = y_1 / e^{c\cdot x_1} = y_2 / e^{c\cdot x_2}$a=y1/ec⋅x1=y2/ec⋅x2

✅ Psst! Given two points, the **exponential function calculator** will do all this math for you!

## FAQ

### What exponential function goes through the points (0, 2) and (1, 4)?

The exponential function these two points lie on is **f(x) = 2·2 ^{x}**. When

*x = 0*, we have

*y = 2·2*, and when

^{0}= 2*x = 1*, we have

*y = 2·2*. Other points on this line are (2, 8), (3, 16), and (4, 32).

^{1}= 4If you want to learn how to find the exponential function from two points, head on over to Omni's exponential function calculator.

### What exponential function goes through the points (0, 4) and (1, 12)?

The exponential function that goes through these two points is **f(x) = 4·3 ^{x}**. Analyzing the points:

- When
*x = 0*, the corresponding y-value is*y = 4·3*.^{0}= 4 - When
*x = 1*, we have*y = 4·3*.^{1}= 12 - If we continue this line, we have more points: (2, 36), (3, 108), and (4, 324).

If you need to find the exponential function from two points, head on over to Omni's exponential function calculator.