Number of solutions to a equation in log

C. exactly two real solutions

\(\log_{3}{x}-\log_{x}{3}=2\)

\(\log_{3}{x}-\frac{1}{\log_{3}{x}}=2\)

Let \(\log_{3}{x}=K\)

\(K-\frac{1}{K}=2\)

\({K}^{2}-2K-1=0\)

\(K=\frac{-(-2)\pm\sqrt{{-2}^{2}-4(1)(-1)}}{2(1)}\)

\(K=\frac{2\pm\sqrt{8}}{2}\)

\(K=\frac{2\pm2\sqrt{2}}{2}\)

\(K=1+\sqrt{2}\)  OR  \(K=1-\sqrt{2}\)

Two real solutions