Contents of this Precalculus episode:
Logarithmic Functions, Logarithmic equations, Exponential identities, Logarithmic identities, solve the exponential equations, Solve the logarithmic equations.
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.
Precalculus episode