Exponents and logarithms

Content of the topic


Exponential functions and identities

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.


Logarithmic functions and identities

Here comes the logarithm

Now a new character enters the scene, the logarithm.

Well, this logarithm is a splendid thing, but requires a bit if an explanation.

All it is, really, that  tells us to which power should “a” be raised in order to get x.

Here is this one, for instance:

This means that we have to find to which power we should raise 2, so that we get 8.

Well, 23=8, so the answer is...

Or let’s see this one:

Well, let's see

Here comes a bit more difficult case:

The question is how 8 would turn into 2. Dividing by 4 is not a good answer here, because we are looking for something exponential.

The correct answer is:

Let’s figure out what this is:

The question is: 8 raised to which power would result in 16?

Well, there is something common in 8 and 16, namely 2, because 23=8 and 24=16.

Then, we get from 8 to 16 by first turning 8 into 2,

and then from 2 we make 16.

After all these, this should not be a problem:

Neither this one:

And now let's see the logarithmic identities.

LOGARITHMIC IDENTITIES

One of the biggest benefits of the logarithm is that we can solve equations like this:

We take the logarithm of both sides.

And Voila!

In general, if we have something like this: 

then we can get x by doing this:

We should remember the reverse of it, too. If

then we can get x by doing this:

Solving exponential equations

Solving logarithmic equations

Let’s solve these, for instance:

And now let's see the functions.

Well, the logarithm is only defined for positive x values.

If the base is greater than 1, then the function is increasing.

If the base is less than 1, then it is decreasing.