Using a calculator x=-0.12776 approx.
However, I think you might have meant 5^(2x-1)-5^(x-1)=0, so 5^(2x-1)=5^(x-1), 2x-1=x-1, in which case x=0. The same answer applies if the base is 6 instead of 5.
Solution using logs and calculator:
5^(2x-1)=6^(x-1)
Take logs of both sides (any base):
(2x-1)log5=(x-1)log6
(2x-1)/(x-1)=log6/log5=1.1133 approx.
2x-1=1.1133(x-1)=1.1133x-1.1133
x(2-1.1133)=1-1.1133=-0.1133
0.8867x=-0.1133, x=-0.1133/0.8867=-0.12776 approx