I just have a quick question regarding this question:

Question

Simplify. State any restrictions on the variables.

log(x^2+7x+12)-log(x^2-9)

my answer is log[(x+4)/(x-3)] which is correct. However, for the restrictions, I believe that they are: x>3 and x<-4.. But the back of the book says only x>3.

When I plug in x<-4 any number such as x=-5 the expression is defined. So.. my question, is the back of the book wrong? Are there 2 restrictions and not one?

Thanks, please explain your answer fully.

Answer 1

remember you can only take logs of positive numbers.

so in log [(x+4)/(x-3)]

x > -4 AND x > 3,
so clearly x > 3

Answer 2

It seems to me that the restriction x>3 is based on
log[(x+4)/(x-3)] = log(x+4) - log(x-3)
which will not permit x≤3, which also covers x≤-4.

The restriction for the simplification is simply x ≠ -3, otherwise we cannot cancel the (x+3) term.

Finally, the function
log[(x+4)/(x-3)]
is defined for
{x ∈ � : x < -4} ∪ {x ∈ � : x > 3}
as you have correctly determined.