In this example, we're going to solve an equation involving logarithms.
Notice here in this equation,
that each term
has the natural log being taken of it.
So what we want to do, is we want to write
this expression here as one term.
Since the two are being added together
this means that it's the product.
So, I can rewrite this as
as the natural log of x - 6
x + 3
At this point, notice that both sides
have the natural log on them.
This natural log of this
should equal the natural log of this.
Since that's the case,
then that must mean that this
must equal this.
So we can take out the natural log
and rewrite our equation as
x - 6 
x + 3
equals 22.
And now we would solve this the way we normally would.
Remember that for this type of problem
we would have to FOIL
the left side.
Put everything on one side, set it equal to 0
and then solve.
So, let's FOIL this x - 6 times x + 3.
FOILing it, we find x squared
minus 3x
minus 18 equals 22.
Place everything on one side and set it equal to 0.
We'll do this by subtracting 22 from both sides.
And now at this point, we need to factor
and solve.
Two numbers that multiply together to give me -40
but add up to give me -3.
This is going to be -8 and positive 5.
So, we have two possibilities.
x is equal to 8
x is equal to negative 5
Now, before we can state
that these are the solutions, we must go back
and check in our problem.
Plug in x = 8.
Remember that this means the product.
So, we can multiply these two together.
And that is true.
Now, let's plug in x = -5.
Now, though this could be true, notice that we have
the natural log of negative numbers.
You cannot take the natural log of negative numbers.
Therefore, negative 5 is not part of the solution.
So, our only answer is x = 8.
