TITLE: Approximating $\frac {\log(5/4)}{\log(3/2)}$ to a rational number
QUESTION [9 upvotes]: I'm making a phone game, and I need to approximate $\frac {\log(5/4)}{\log(3/2)}$ to a rational number $p/q$.
I wish $p$ and $q$ small enough. For example, I don't want $p$, $q\approx 10^7$; it's way too much for my code.   
In the game, there's two way to upgrade ability. Type A gives an additional $50\%$ increase at once. and type B gives $25\%$.
What I want to know is how many times of upgrade $(x,y)$ provides the same additional increase. So what I've done is solve $(3/2)^x = (5/4)^y$  respect to $\frac xy$.   
Can you provide me a way to construct sequence $p_n$, $q_n$ which approximate the real number?
Thank you in advance.

REPLY [2 votes]: Use the following equation 
$$\log(1+\frac{1}{x})\approx\frac{3+6x}{6x^2+6x+1}$$
so
$$\log(\frac{5}{4})=\log(1+\frac{1}{4})$$
$$\log(\frac{3}{2})=\log(1+\frac{1}{2})$$
the approximated value will be 
$$\frac {\log(5/4)}{\log(3/2)}\approx \frac{333}{605}=0.55041322$$