Solve the following equation for $$ x $$ : $$ \log _{x} 81=2 $$ $$ x=\square $$

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Solve the equation $$ \log_{x}81=2 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\log_{x}{\left(81ight)}=2$$
- step1: Find the domain:
  $$\log_{x}{\left(81ight)}=2,x \in \left(0,1ight)\cup \left(1,+\inftyight)$$
- step2: Convert the logarithm into exponential form:
  $$81=x^{2}$$
- step3: Swap the sides:
  $$x^{2}=81$$
- step4: Simplify the expression:
  $$x=\pm \sqrt{81}$$
- step5: Simplify:
  $$x=\pm 9$$
- step6: Separate into possible cases:
  $$\begin{align}&x=9\&x=-9\end{align}$$
- step7: Check if the solution is in the defined range:
  $$\begin{align}&x=9\&x=-9\end{align},x \in \left(0,1ight)\cup \left(1,+\inftyight)$$
- step8: Find the intersection:
  $$x=9$$
The solution to the equation $$ \log_{x} 81 = 2 $$ is $$ x = 9 $$.
$$ x = 9 $$

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