Calculate the limit? 
The answer sheet says:

I understand all of this apaprt from the last bit where it says 
I don't understand how/why they have put logs and exponentials in there. 

Since $exp$ and $log$ are reciprocals, then
\begin{equation*}
a^{b}=e^{\ln (a^{b})}=e^{b\ln a}.
\end{equation*}
So,
\begin{equation*}
\left( \frac{3}{4}\right) ^{x}=e^{x\ln \left( \frac{3}{4}\right) }=\frac{1}{%
e^{-x\ln \left( \frac{3}{4}\right) }}=\frac{1}{e^{x\ln \left( \frac{4}{3}%
\right) }}.
\end{equation*}
On the other hand, $\ln \left( \frac{4}{3}\right) >0$ since $\ln x>0$ for
all $x>1,$ and $\frac{4}{3}>1.$
