Question

ln (x-2) +ln (2x-3)=2ln x

ln(x-2)(2x-3)=ln x^2
ln 2x^2-7x+6=ln x^2

is as far as I got.

Answer 1

$D:x-2\ \textgreater \ 0 \wedge 2x-3\ \textgreater \ 0 \wedge x\ \textgreater \ 0\\ D:x\ \textgreater \ 2 \wedge x\ \textgreater \ \dfrac{3}{2}\wedge x\ \textgreater \ 0\\ D:x\ \textgreater \ 2\\\\ \ln (x-2) +\ln (2x-3)=2\ln x \\ \ln (x-2)(2x-3)=\ln x^2\\ (x-2)(2x-3)=x^2\\ 2x^2-3x-4x+6=x^2\\ x^2-7x+6=0\\ x^2-x-6x+6=0\\ x(x-1)-6(x-1)=0\\ (x-6)(x-1)=0\\ x=6 \vee x=1\\ 1\not \in D\\ \boxed{x=6}$