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.