I'm working through Courant's book on Introductory Calculus and Analysis and I was reading his proof of the Additivity of Logarithms and the reasoning for when $\ 0 \leq x \leq 1 $ didn't seem sound so I thought someone could let me know if this is allowed.
He starts with
$\ \log{x} + \log{y} = \log{x} + \log{\frac{x}xy} $
and then reduces this to
$\ \log{x} + log{\frac{1}x} + \log{xy} $.
For this step then, doesn't he use the additive theorem of logarithms, the one we are currently proving. This seems circular and I wanted anyone to clarify. Why or why not this is allowed?
As a side note, he also produces another proof of this theorem shortly after, starting from the formula of Riemann sums of a $ \frac{1}x $, which I like better, but if the proof above is in the book, it must be considered sufficient as well.
Best Answer
The first part shows that if $a > 1$ and $b >0$, then $$ \log(ab) = \log a + \log b. $$ I use different letters on purpose. Then he uses this for $a = 1/x > 1$ and $b = xy > 0$.
As a side note, any proof of this result (or anything else) using Riemann sums will necessarily be tedious. The easiest way is to apply the fundamental theorem of calculus and the chain rule to $f(x) = \log(ax)$, where $a > 0$ is fixed.