Es gelten folgende Eigenschaften: $$\log a^x =x\cdot \log a \\ \log a+\log b =\log a\cdot b \\ \log a - \log b =\log \frac{a}{b}$$
Wir bekommen also folgendes: $$\frac{\ln a-\ln (2a^2)}{\ln (a^3)-\ln (4a^4)} \\ =\frac{\ln a-\left(\ln (2)+\ln (a^2)\right)}{\ln (a^3)-\left(\ln (4)+\ln (a^4)\right)} \\ =\frac{\ln a-\left(\ln (2)+2\ln (a)\right)}{3\ln (a)-\left(\ln (4)+4\ln (a)\right)} \\ =\frac{\ln a-\ln (2)-2\ln (a)}{3\ln (a)-\ln (4)-4\ln (a)} \\ =\frac{-\ln (2)-\ln (a)}{-\ln (4)-\ln (a)} \\ =\frac{\ln (2)+\ln (a)}{\ln (4)+\ln (a)} \\ =\frac{\ln (2a)}{\ln (4a)}$$
