Answer: \(x=\frac83\).

Answer: \(x=\frac83\).

Use the identity

\(\log_x3=\frac1{\log_3x}.\)

Therefore

\(\frac{2}{\log_x3}=2\log_3x.\)

The equation becomes

\(\log_3(11x-8)=1+2\log_3x.\)

Since \(1=\log_33\), the right-hand side is

\(\log_33+ \log_3x^2=\log_3(3x^2).\)

Hence

\(\log_3(11x-8)=\log_3(3x^2).\)

So

\(11x-8=3x^2.\)

Rearranging gives

\(3x^2-11x+8=0.\)

Factorising,

\((3x-8)(x-1)=0.\)

Thus \(x=\frac83\) or \(x=1\). Since the question gives \(x\gt 1\), and \(\log_x3\) is not defined for \(x=1\), the solution is

\(x=\frac83.\)