Logarithm equation (solving for $x$)

Question

I am stuck on this problem. Find the value of $x$ in this expression:

$$ \ln(x-2) \cdot \ln(x)=\ln(64) $$

Answer 1

Little bit of algebra that is not interesting gives $(x-2)^{\log{x}}=64$ or $x^{\log{x-2}}=64$. RHS is strictly increasing in $x$, equals $1$ at $x=3$ and is larger than $70$ at $x=9$. Hence the unique solution is in $[3,9]$. Numerically, it is approximately $8.78151$.