Answer
(a) $2\ln\sqrt e=1$
(b) $\ln(\ln e^e)=1$
(c) $\ln e^{-x^2-y^2}=-x^2-y^2$
Work Step by Step
This whole exercise is based on this property: $$\ln e^x=x\hspace{1cm}x\gt0$$
(a) $$2\ln\sqrt e$$
- First, apply Power Rule: $$2\ln\sqrt e=\ln(\sqrt e)^2=\ln e=1$$
Therefore, $$2\ln\sqrt e=1$$
(b) $$\ln(\ln e^e)$$
- Considering $\ln e^e$ first: $$\ln e^e=e$$
Therefore, $$\ln(\ln e^e)=\ln e=1$$
So, $$\ln(\ln e^e)=1$$
(c) $$\ln e^{-x^2-y^2}=-x^2-y^2$$
