What is the range of f(x) = log(x – 3)?
The range of any standard log function like yours is the set of real numbers.
That is, choosing the appropriate value of x will yield any y value you want
suppose we want y or f(x) to be -500
-500 = log(x-3)
x-3 = 10^-500
x = 3 + 10^-500 , a number just a bit to the right of +3
(the closer you get to +3 from the right, the larger into the negatives the y will become)
try the same thing for large values of x
e.g. x = 1 000 000 000
y = log (1 000 000 000 - 3) = 8.999..
to get really large values of y as an answer your x must be un-measurably large.