As far as I know logs with arbitrary bases are not implemented in WeBWorK. I suspect that for someone who has a better understanding of MathObjects it wouldn't be hard to define a new function of two arguments, "log_", which evaluates log_(x)(y) to log(x)/log(y).
In the meantime you can always ask your question in the form (x-2)^3=z becomes log_(a)(b)=c. What are a, b and c?