Is Log(x) Continuous on the Interval (0, ∞)?

[Yagoda]

Homework Statement

Prove that f(x)=\log x is continuous on (0, \infty) using that
(1) f is continuous at x=1 and
(2) \log(xy) = \log(x) + \log(y)

Homework Equations

The definition of continuity: for all \epsilon >0, there exists a \delta>0 such that if |x-x_0| < \delta then |f(x) - f(x_0)| < \epsilon.

The Attempt at a Solution

I think I've figured out how to do this using a more standard epsilon-delta proof, but it doesn't really make use of the two facts.
From what I can tell, it seems like you trying to be able to use the continuity at x=1 to "slide" the continuity down to 0 and up to infinity, but I'm not sure how to do this in a valid way. The only way I've managed to use fact 2 is rewrite things like \log x = \log(x \times 1) = \log(x)+\log(1) = \log(x), which hasn't gotten me very far.

 

[Axiomer]

Try replacing x_0 with xy for some y.

 

[verty]

Look at $f(x) - f(x_0)$, this becomes $log(x) - log(x_0)$.

 

[lurflurf]

hint
log(x+h)-log(x)=log(1+h/x)-log(1)