Arguments of logarithms, no matter what the base, must not be zero or negative.
Denominator.
Let's find what x value makes the denominator zero:
x - 6 = 0
Add 6 to both sides:
x = 6
This is a value we cannot allow for x. It must be excluded from the domain.
Argument of the logarithm.
We want a positive argument:
There are different ways to solve this. One way is to subtract the terms on the left. Of course we need common denominators first:
Subtract (Be careful when subtracting!) :
This says we have a positive fraction. When do we get positive fractions? Answer: When the numerator and denominator are both positive or when they are both negative. This idea, which you probably knew well, can be expressed in the algebraic form:
( and ) or ( and )
(Look at this and see if you can understand how this says "both positive or both negative".)
Now we solve this compound inequality.
( and ) or ( and )
In the first pair, in order for x to be greater than 17 and 6, it would have to be greater than 17. In the last pair, in order for x to be less than 17 and 6, it would have to be less than 6. So
( and ) or ( and )
simplifies to or
Since or excludes the number which makes the denominator zero, this is our domain.
(