Inequalities with both x and y

Question

If I had an equation like $b^x>$ $d^y$ where $b,d $ are real, and wanted to find out when it is true, for finitely many $+ve$ integer values of $x,y$, I would start by writing $z = b^x$ and $z=d^y$ then on separate graphs draw them, first one $z$ against $x$, second one $z$ against $y$, then see if one graph is always greater than the other one for some finite range of $x$ and $y$ values (I fixed $0<b<1$ and $d>1$, however doing this approach gets me the wrong answer, as it is clear that $a^x$ is never greater than $d^y$ for $+ve$ integers $x,y$, however if I plot $y=b^x$ and then $x=d^y$ I get a finite region where $b^x$ is greater. Which approach is correct? In the picture you can see the first graph is never above the second one for positive $x,y$.