$ log_{b}^{N^2}=log_{a}^{N}+ log_{c}{N} \Rightarrow \frac{1}{log_{N^2}^{b}}=\frac{1}{log_{N}^{a}}+\frac{1}{log_{N}^{c}}= \frac{log_{N}^{ac}}{log_{N}^{a}log_{N}^{c}} \Rightarrow \frac{2}{log_{N}^{b}}=\frac{log_{N}^{ac}}{log_{N}^{a}log_{N}^{c}} $
$$\color{blue}{ \Rightarrow \frac{2log_{N}^{a}}{log_{N}^{b}}=\frac{log_{N}^{ac}}{log_{N}^{c}} \Rightarrow 2log_{b}^{a}=log_{c}^{ac} \Rightarrow log_{b}^{a}= \frac{1}{2} log_{c}^{ac}} $$
$$\Rightarrow log_{b}^{a}=log_{c^2}^{ac} \Rightarrow log_{a}^{b}=log_{ac}^{c^2}$$
$$ \Rightarrow\color{red}{(ac)^{log_{a}^{b}}=c^2} $$
