$$ y=\ln((3x-2)^2) $$
State the domain and the coordinates of the point where the curve crosses the x-axis
At first sight, you say that the domain is $x>\frac23$ because $\ln$ is undefined for negative numbers, so you just rearrange $3x-2>0$.
But the input of $\ln$ is squared, which means there are 2 roots, namely $1$ and $\frac13$.
By the law of logarithms $$ \ln(x^2)=2\ln(x) $$ Therefore, the function for $y$ can be rewritten as $$ y=2\ln(3x-2) $$ The problem is that half the graph disappears. Now that the input isn't squared, $y$ is undefined for $x\le\frac23$ ($3x-2$ becomes negative) and the entire left half is gone.
So what's the answer? How many roots are there? It seems that math is contradicting itself.