# Proving Log(x) is Continuous

## 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.

Try replacing $x_0$ with xy for some y.

Last edited:
• 1 person
verty
Homework Helper
Look at ##f(x) - f(x_0)##, this becomes ##log(x) - log(x_0)##.

lurflurf
Homework Helper
hint
log(x+h)-log(x)=log(1+h/x)-log(1)