I have the following formula:
$$\frac{\ln(\frac{a}{x})}{k} + \frac{\ln(b) - \ln(x)}{k}$$
where $k>0$, $a>0$, but $b<0$ and $x<0$.
I cannot use the quotient rule for logarithms to change $\dfrac{a}{x}$ to $ln(a) - ln(x)$ since I am dealing with a negative and a positive.
But does the quotient rule allow me to change $\ln(b) - \ln(x)$ to $\dfrac{b}{x}$ since here I am dealing with the logarithm of two negative numbers, and hence dealing entirely in complex space?
It appears Mathematica concurs, but I am uneasy about making the change without human confirmation.
For example, in Mathematica
m = -9;
n = -27;
FullSimplify[Log[m/n] - (Log[m] - Log[n])]
generates the answer 0, and perhaps not surprisingly WolframAlpha concurs:
