Exponential Function Calculator

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 $x$ or $y$ 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.

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:

  1. Decide the function formula shape (e.g., $b^x$ or $p\cdot e^{kx}$).

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

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

• When in "evaluate the function" mode:

  1. Enter the exponential function's parameters.

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

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.

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

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:

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

• $f(x) = b^x$
• $f(x) = a \cdot b^x$
• $f(x) = e^{kx}$
• $f(x) = p \cdot e^{kx}$

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

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 $(x_1, y_1)$ and $(x_2, y_2)$, you can easily solve for $a$ and $b$ — in other words, you can solve:

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

and from there, we can determine that:

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

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

What if the base is $e$?

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

Here's how:

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

The exponential function these two points lie on is f(x) = 2·2^x. When x = 0, we have y = 2·2⁰ = 2, and when x = 1, we have y = 2·2¹ = 4. Other points on this line are (2, 8), (3, 16), and (4, 32).

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

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

1. When x = 0, the corresponding y-value is y = 4·3⁰ = 1.
2. When x = 1, we have y = 4·3¹ = 12.
3. 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.