Answer
$\log_a{\left(\frac{1}{N}\right)}=\log_a{\left(N^{-1}\right)}=-1\cdot\log_a{N}=-\log_a{N}, a\ne1$
Work Step by Step
Recall:
$(1)\quad \quad\dfrac{1}{N}=N^{-1}\\
(2) \quad \log_a{x^{m}}=m\log_a{x} $
Work on the LHS and use rule $(1)$ in the recall part above to obtain:
$$
\begin{aligned} \log _{a}\left(\frac{1}{N}\right)&= \log _{a}\left(N^{-1}\right)\\
\\\text{Use rule (2) in the recall part above to obtain:}
\\\log _{a}\left(\frac{1}{N}\right)&=-1 \log _{a} N \\=&-\log _{a} N \\=& R H S \end{aligned}
$$
$$\therefore LHS=RHS$$
