If

${\displaystyle {\left(\frac{5}{4}\right)}^m\approx {\left(\frac{2}{1}\right)}^n}$

then

${\displaystyle m\log \frac{5}{4}\approx n\log 2}$

and we shall find

${\displaystyle \frac{m}{n}\approx \frac{\log 2}{\log 5/4}.}$

Using Euclid's algorithm the result can be represented as a continued fraction.

${\displaystyle \frac{\log 2}{\log 5/4}=3+\frac{Q}{\log 5/4}}$

where $0<\frac{Q}{\log 5/4}<1$. So

${\displaystyle \frac{\log 2}{\log 5/4}=3+\frac{1}{\frac{\log 5/4}{Q}}=3+\frac{1}{9.4087788}=3+\frac{1}{9+\frac{1}{2.4463109}}}$

giving the first approximation $\frac{28}{9}$. Repeating the algorithm:

${\displaystyle \frac{\log 2}{\log 5/4}=3+\frac{1}{9+\frac{1}{2+\frac{1}{2+\frac{1}{4+...}}}}}$

Truncating this continued fraction gives the successive approximations $\frac{59}{19}$, $\frac{146}{47}$ and $\frac{643}{207}$.