Down-to-Earth Explanation of Exponential Function -

The exponential function is frequently used in Calculus. I’ll explain what this function is like in this post.

Recall that you can write powers of two like this (we use $ \cdot $ for multiplication):

$2^{-3} = (1/2) \cdot (1/2) \cdot (1/2) = 1/8 $

$2^{-2} = (1/2) \cdot (1/2) = 1/4 $

$2^{-1} = 1/2 $

$2^0 = 1$

$2^1 = 2$

$2^2 = 2 \cdot 2 = 4$

$2^3 = 2 \cdot 2 \cdot 2 = 8$

You can also write out powers of ten:

$10^{-3} = (1/10) \cdot (1/10) \cdot (1/10) = 1/1000 = 0.001 $

$10^{-2} = (1/10) \cdot (1/10) = 1/100 = 0.01 $

$10^{-1} = 1/10 $

$10^0 = 1$

$10^1 = 2$

$10^2 = 10 \cdot 10 = 100$

$10^3 = 10 \cdot 10 \cdot 10 = 1000$

In fact, you can try this out with many different numbers. One such number is the following.

$$ e = 2.7182 \dots $$

This number is known as $ e $. We can also compute powers of that number:

$e^{-3} = (1/e) \cdot (1/e) \cdot (1/e) = 0.0497 \dots$

$e^{-2} = (1/e) \cdot (1/e) = 0.1353 \dots $

$e^{-1} = 1/e = 0.3678 \dots$

$e^0 = 1$

$e^1 = 2.7182 \dots$

$e^2 = e \cdot e = 7.3890 \dots$

$e^3 = e \cdot e \cdot e = 20.0855 \dots$

So, what is the exponential function? The exponential function is the following: $$ y = e^x $$ It is plotted below.

The points labelled in the above graph are $(-3, e^{-3})$, $(-2, e^{-2})$, $(-1, e^{-1})$, $(0, 1)$, $(1, e)$, $(2, e^2)$, and $(3, e^3)$.

There are some things to note about this function:

The exponential function increases. As $ x $ gets larger, $ e^x $ gets much larger.
The number $ e^x $ is always positive.
When $ x = 0 $, $ e^x = 1 $.

Down-to-Earth Explanation of Exponential Function