# The Exponential Function as an Isomorphism

- Author:
- Brad Ballinger

- Topic:
- Exponential Functions, Logarithm

## How to play with this applet

1) Mess around. See what you can figure out on your own.
2) Reset the applet and look for the following:
a. The left number line includes positive and negative numbers.
b. The right number line consists of only positive numbers.
c. As you move either blue point, which parts of the exponential expression (green) change? Which parts stay the same?
d. As you move either blue point, which parts of the logarithmic expression (purple) change? Which parts stay the same?
e. Move the left point to the "additive identity", 0. What does it match to on the right?
f. With all the boxes checked, verify in a few examples that *adding* numbers on the left number line corresponds to *multiplying* numbers on the right number line. This can be summarized by saying: "An exponential function turns addition into multiplication, while a logarithm turns multiplication into addition."
It's true that any exponential function y=b^x (where b>0 and b isn't 1) is a "correspondence" (a.k.a. "bijection", a.k.a. "one-to-one and onto map") between the set of all real numbers and the set of positive real numbers; however, I hope that by playing with this applet you find that there's something deeper going on.