TITLE: I need to prove the continuity of $f(x)=\log x$ using a $\epsilon-\delta$ proof
QUESTION [9 upvotes]: I need to prove the  continuity of $f(x)=\log x$ using a $\epsilon-\delta$ proof
These is what I have so far but am not sure how to continue
$|\log x-\log a| < \epsilon$ 
$\log a- \epsilon < \log x < \log a+ \epsilon$
$\frac{a}{e^\epsilon} < x < {a}e^\epsilon$
Any help is appreciated

REPLY [9 votes]: By your inequality, the absolute value of the difference is $\lt \epsilon$ if 
$$\frac{a}{e^{\epsilon}}-a \lt x-a\lt ae^\epsilon -a$$
(we subtracted $a$ from each side of each of your two inequalities).
Let $\delta=a\min\left(1-\frac{1}{e^{\epsilon}}, e^\epsilon -1\right)$. 
Remark: Actually, $1-\frac{1}{e^{\epsilon}}$ is the smaller of the two, so in effect we are letting that be $\delta$. But we really don't need to bother finding that out: all we need to do is to show there is a $\delta$ that works.