solve for x : log base 9 X^2=9

  $$\small$\text{ Solve for x : $\log_9{(x^2)}=9$ }}$$

$$\small{\text{Formula: $\boxed{~ \log_b{(b^a)}=a~}$}}\\\\ \small{\text{ We have $b=9$, and $a = 9$ so $\log_9{(9^9)}=9$, but $9^9$ is $x^2$ }}\\\\ \small{\text{$ \begin{array}{rcl} x^2 &=& 9^9\\ x &=& 9^{\frac{9}{2}}\\ x &=& \left(\sqrt{9} \right)^9\\ x &=& \left(\sqrt{3^2} \right)^9\\ x &=& 3^9\\ \mathbf{x} & \mathbf{=} & \mathbf{19683}\\ \\ \hline \\ \end{array} $}}\\$$

$$\\log_9x^2=9\\\\ 2log_9x=9\\\\ log_9x=9/2\\\\ 9^{log_9x}=9^{9/2}\\\\ x=9^{9/2}\\\\ x=3^9\\\\ x=19683$$

check:

$${{log}}_{{\mathtt{9}}}{\left({{\mathtt{19\,683}}}^{{\mathtt{2}}}\right)} = {\mathtt{9}}$$