Problem: Find the solution that contains a logarithmic term to the nonlinear PDE for $x \in (0, \pi)$, $t > 0$, $ t u_t - x u_x - (3 - 2x \cot(x) ) u \log u = 0, \quad u(0, t) = 1, u(\pi, t) = 1, u(x, 0) = 1 $. Please place your final solution in a $\boxed{}$ LaTeX Environment. The only remaining parameters in your final solution should be $x$ and $t$; do not define additional parameters or constants. There should be no words inside the $\boxed{}$ environment; only an expression that can be evaluated by subsituting in values of $x$ and $t$.
Solution: $\boxed{ u(x,t) = \exp(\sin^2(x) t^3 \log(1 + xt)) }$