How can one maximize the ratio of two logarithms $ \frac{\log{f(x)}}{\log{g(x)}}$ where the argument to each logarithm is the (positive) ratio of two first-degree polynomials? I have tried differentiating and solving for $ x $ but became caught in a quagmire of logarithms. Are there any simplifications or transformations that can be applied that will preserve maxima or zeros of the derivative?
For discussion, here's a complete example - selecting the amount of fuel to carry to maximize the stage specific impulse of an ideal rocket described by the Tsiolkovsky rocket equation.
$$ \frac{\log{\frac{(\alpha+1)x + \beta}{\alpha x+\beta}}}{\log{\frac{(\alpha + 1)x+\beta}{\beta-\gamma}}} $$
$ \alpha $, $ \beta $, and $ \beta - \gamma $ are all positive. $ \alpha $ is the mass of storage to contain one unit of fuel, $ \beta $ is the combined mass of the payload and rocket engine, and $ \gamma $ is the mass of the rocket engine alone.
I'm not sure this really answers the intended question; I'm just leaving it here for reference.
The derivative of $\log f(x)/\log g(x)$ is zero when $$\frac{f'(x)}{f(x)\log f(x)}=\frac{g'(x)}{g(x)\log g(x)}.$$ When you have $f(x)=a(x)/b(x)$ and $g(x)=c(x)/d(x)$, this becomes $$\frac{a'(x)/a(x) - b'(x)/b(x)}{\log a(x) - \log b(x)}=\frac{c'(x)/c(x) - d'(x)/d(x)}{\log c(x) - \log d(x)}.$$ When $a(x)$, $b(x)$, $c(x)$, and $d(x)$ are first-degree polynomials, the derivatives in the above expression become constants, but the expression doesn't simplify much after that.