Given a positive integer $n$, we can say that
i). $2^x \le n < 2^{x+1}$
ii). $3^y \le n < 3^{y+1}$
iii). $5^z \le n < 5^{z+1}.$
By using the first two inequalities, can we find or give an inequality for $z$ in terms of $x$ and $y$?
Edit. I can get the inequalities by taking logarithms like
$$ z < \frac{\ln 2}{\ln 5} (x+1) \text{ or }< \frac{\ln 2}{\ln 5} x < z+1 .$$ but I rather want something about the integerness properties of $z$. For instance, by knowing the values of $x$ or $y$ modulo some number $m$, can we say something about $z \pmod m$?