Take logs of both sides:
log(23x+1)=log(3x-2),
(3x+1)log(2)=(x-2)log(3),
3xlog(2)+log(2)=xlog(3)-2log(3),
x(3log(2)-log(3))=-log(2)-2log(3),
x=(-log(2)-log(9))/(log(8)-log(3)),
x=-log(18)/log(8/3)=-2.9469 approx. (The base of the logs doesn't matter as long as both logs are to the same base.)