Contents of this Precalculus episode:

Logarithmic Functions, Logarithmic equations, Exponential identities, Logarithmic identities, solve the exponential equations, Solve the logarithmic equations.

Text of slideshow

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.


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.


Logarithmic functions and identities

Enter the world of simple math.
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