Let’s discuss the exponential functions by starting with the basics, the exponential identities.

Exponentials are fun, and for starters, it is enough to know that

but don’t panic, there won’t be anything devilish here.

The first exponential identity tells us what happens if we multiply this by - let’s say - 62.

Well, let's check it out.

If we multiply these, then

the exponents will be added up.

This is the first identity.

EXPONENTIAL IDENTITIES

Let's see what happens if we divide these.

But there is a minor inconvenience.

Here comes.

Well, when the exponent of the denominator is greater, then the result is a fraction.

Here the exponent will be negative.

Next, let's see how to raise a power to a power.

Well, like this:

We have to multiply the exponents.

Then, we have this thing:

So, what could this be?

Let's see what happens if we apply our newest identity to this.

So, it is something whose square is 9.

There happens to be such a thing, we call it .

So, a fractional exponent means a radical.

Developing the previous two identities a bit further, we get a third one.

If we have something like this:

then we can try applying this formula.

Here come a few more formulae,

but now, finally, let’s see the functions.

The function of 2x looks like this: This one is 3x.

If the base number is between 2 and 3, then the function will be between 2x and 3x.

For example, one such number is

2.71828182845904523536028747135266249775724709369995…

This number has a powerful significance in mathematics, and to simplify things, it is called “e”.

Thus this function is ex.

All exponential functions with a base greater than 1 look something like this.

If the base is less than 1, well, that is an entirely different animal.