TITLE: How to compare logarithms $\log_4 5$ and $\log_5 6$?
QUESTION [11 upvotes]: I need to compare $\log_4 5$ and $\log_5 6$. 
I can estimate both numbers like $1.16$ and $1.11$. Then I took smallest fraction $\frac{8}{7}$ which is greater than $1.11$ and smaller than $1.16$ and proove two inequalities: $$\log_4 5 > \frac{8}{7}$$
$$\frac{7}{8}\log_4 5 > 1$$
$$\log_{4^8} 5^7 > 1$$
$$\log_{65536} 78125 > 1$$
and
$$\log_5 6 < \frac{8}{7}$$
$$\frac{7}{8}\log_5 6 < 1$$
$$\log_{5^8} 6^7 < 1$$
$$\log_{390625} 279936 < 1$$
thats why I have $\log_5 6 < \frac{8}{7} < \log_4 5$.
But for proving I need estimation both logarithms (without this estimation I cannot find the fraction for comparing).
Can you help me to find more clear solution (without graphs)

REPLY [3 votes]: $$\frac54>\frac65\land 4<5\implies\frac{\log\dfrac54}{\log 4}>\frac{\log\dfrac65}{\log 5}\implies\frac{\log5}{\log 4}>\frac{\log6}{\log 5}.$$