TITLE: $\log_9 71$ or $\log_8 61$
QUESTION [16 upvotes]: I am trying to know  which one is bigger :$$\log_9 71$$ or  $$\log_8  61$$  how can i know without using a calculator ?

REPLY [16 votes]: Ok, so we have that $$\frac{61}{64}>\frac{71}{81} \implies \log_{8}\left(\frac{61}{64}\right)>\log_{9}\left(\frac{71}{81}\right)\cdot\log_{8}(9)>\log_{9}\left(\frac{71}{81}\right)$$
By application of the change of base formula and the fact that $\log_{8}(9)>1$, which is trivial.