I teach the following equation as an example of one that has no closed-form analytic solution in my calculus class. It comes up in biology.
$$ a^{1 - x/b} (b - x log(a)) = b $$
Interestingly, I recently tried solving it numerically for fun. When I did this for different values of $b$, I found that the solution as a proportion of $b$ remained completely unchanged, no matter what $b$ was. Is there any way to prove this property? That is, I don't have an expression for $x^*$ which is a solution of the above equation, but even without this expression, can I prove that $x^*/b$ is independent of $b$ or $x^*$ is directly proportional to $b$? The example is illustrative, it would be useful to think of strategies to do this for other examples too.
Plugging $x'=rb$, for some real fixed $r$, your equation becomes $$a^{1-r}(b-b\,r\, \log{a})=b \iff a^{1-r}(1-r\, \log{a})=1$$ As you can see, $r$ does not depend on $b$, which I think gives you your answer.