The limits on the domain of log functions come from the fact that is not possible to take the log of a negative number. Log functions will not have limits on the range.
First, recall the graph of the "parent function" for y = log x.
To refresh your memory, the parent graph for log x has an anchor point at (1,0); from there it asymptotes downward to the left approaching x = 0; from the point, it ascends gradually to the right with no upward limit. If you need to see the parent graph, plug y = log x into a graphing utility or website.
If you prefer, you can build the parent function graph from scratch by plotting points for y = log x.
Once you visualize the parent function, it is easy to tell the domain and range. "Domain" is "everything x can be." So the domain of the parent function is greater than x and all the way to positive infinity.
Domain is 0 > x > ∞.
"Range" is "everything y can be." On the left side, the graph goes down to negative infinity. On the right side, it gradually continues to ascend... all the way to positive infinity eventually. So the range is all real number from positive infinity to negative infinity.
Range is -∞ > y > ∞.
Now for the range of any log function, simply consider the way the parent function and graph change as constants are introduced. Using the form...
y = a log(x-h) + k
with possible constant numbers for a, for h, and for k...
The anchor point of (1,0) shifts to (h,k). The h is your horizontal shift, and the k is your vertical shift. The key with log functions is that the asymptote also shifts with the value of h, which changes the domain. Since the range is all real number, the value of a and of k do not affect the domain or range.
Example:
y = log(x-2)
The h value of +2 shifts the parent function and asymptote two units to the right, changing the domain to 2 > x > ∞. The range is still all real numbers from negative infinity to positive infinity.
Example
y = 4 log(x-2) + 7
The stretch of the "a" value at 4 and the upward shift of 7 units do not affect the range or domain, so the domain and range are the same as the previous example.
Now... let's take it one more step, There is one more constant which may affect the log function. I will add it to the format above as the constant b.
y = a log(b(x-h)) + k
The number for b changes the anchor point, but it does not change the domain or range, unless b happens to be negative. If b has a negative value, the graph will be reflected over the asymptote. The range would be unchanged. However, the domain will change. Instead of going from the asymptote to positive infinity, the domain will now be from negative infinity over to the asymptote (non-inclusive).
Example:
y = log(-x)
Since the parent function is reflected over the asymptote, the anchor point will be (-1,0). From there, the function will asymptote downward to the right toward the y-axis (x = 0). From the anchor point, the function will ascend gradually to the left for infinity. So the domain is from negative infinity up to 0.
Domain: -∞ > x > 0.
The range continues to be all real numbers.
Range: -∞ > y > ∞
Note on notation:
If you use interval notation, your domains and ranges will look like this:
D: (-∞,0)
R: (-∞,∞)
using rounded parenthesis rather than brackets since the 0 is not included and since infinity can never be reached.
