TITLE: how do I compare those two logs?
QUESTION [5 upvotes]: I am having trouble how to show which is larger.
$\log4\cdot \log3 \quad\text{or}\quad \log4 - \log3$
base is $10$ for every $\log$.
I really appreciate your help

REPLY [6 votes]: Note that:
$$\begin{align}\log4\cdot \log3 \quad&\text{or}\quad \log4 - \log3 \iff \\
\log 3^{\log 4} \quad &\text{or} \quad\log \frac43 \iff \\
3^{\log 4+1} \quad &\text{or} \quad 4 \iff \\
3^{\log 4+1} \quad >3^{1/2+1} =\sqrt{27}\quad > 5 \quad &> \quad 4.\end{align}$$