After playing around with logarithms, I've found the following coincidences:
$\log_{10}{2} \approx 0.3$, since $2^{10} \approx 10^3$, and $\log_{10}{5} \approx 0.7$, since $5^{1000} \approx 10^{700}$.
I'm sure these are well known. I was just wondering if there was a method or algorithm to generalize these "coincidences" to any base or number. I only have a basic understanding of number theory and would like to learn more.
For example: for $2^b\approx 3^d$, then $\frac{b}{d}\approx\frac{\ln(3)}{\ln(2)}$. So we can find a rational approximant (using continued fractions?) for $\frac{\ln(3)}{\ln(2)}$, say $$\displaystyle{{{\frac{268167867796283}{169195086744492}}}}$$ and $$2^{268167867796283}\approx 3^{169195086744492}$$ and $$\log_2(3)\approx\frac{268167867796283}{169195086744492}$$