Which one is greater: $\log_{13}160$ or $\log_{17}291$?
Comparing logarithms with equal bases is fairly easy. Here they aren't equal, though. In similar problems I have seen that we can compare each of the logarithms with zero or $1$, but here neither of them works, e.g $$\log_{13}160>1=\log_{13}13\\\log_{17}291>1=\log_{17}17$$ What else can we do? P. P. I am not very fluent with functions.
Best Answer
$$\log_{17}291 > \log_{17}289 = 2$$ $$\log_{17}291 > 2$$ $$\log_{13}160 < \log_{13}169 = 2$$ $$\log_{13}160 < 2$$
Thus: $$\log_{13}160 < 2 < \log_{17}291$$ $$\log_{13}160 < \log_{17}291$$