Exponential Function Calculator (2024)

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

💡 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!

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!

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

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:

$$\footnotesize\begin{split}y_1 &= a\cdot b^{x_1} \\y_2 &= a\cdot b^{x_2} \\\end{split}$$y1​y2​​=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⋅bx2​a⋅bx1​​b(x1​−x2​)​=y2​y1​​=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?

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

Here's how:

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