How to compute the utility function when risk aversion is equal to 1?
Take the limit as $\sigma \rightarrow 1$,
\begin{align*}
\lim_{\sigma \rightarrow 1} \frac{C^{1-\sigma}}{1-\sigma}.
\end{align*}
This will then results in log utility. Calculating the limit goes as follows.
Since plugging in $\sigma = 0$ would give $\frac 0 0$, we use L'Hopital's rule.
This gives us
\begin{align*}
\lim_{\sigma \rightarrow 1} \frac{C^{1-\sigma}}{1-\sigma}
  &amp;= \lim_{\sigma \rightarrow 1} \frac{- C^{1-\sigma} \log C}{-1} \\
  &amp;= \log C.
\end{align*}