Proving functions are non-decreasing

I find the concept that recursive functions can be proven to be non-decreasing. Now, I'm wondering: What about non-recursive functions? For example, how would one go about proving that log, or square root, or exponentiation functions are non-decreasing? Do we simply accept that as axiomatic? For example, I know that log(1) is zero in every base and that log(base) is 1 for every base. What about values between 1 and base? I'd like to prove that they are non-decreasing, that is, for any value v between 0 and base, 0 <= log(v) <= log(base). Then I'd like to be able to show that this is true for any v > 0. That is if I have 0 = < v1 <= v2 for any real numbers v1 and v2, how can I prove that:

0 <= log(v1) <= log(v2)

What about other functions, like roots or powers or in general any polynomial with real powers? How could I prove that, for all real numbers r s.t. 0 <=r < 1, the root function on those numbers is non-decreasing for any real root a. (e.g. the square root of 1/4 is 1/2 which is less than the cube root of 1/4 (approx 0.63) or that for reals >= 1, the function is non-increasing?

I'd just like to be able to justify statements like "sqrt function is non-decreasing for values >= 1) or "log function is non-decreasing for values > 0." I suppose that there are tons of other similar functions that fall into the same category as well (though not sine or cosine, of course, except with appropriate bounds).