Not sure how to read this.
If it's log to base 5 of x and of (x-1), the left hand side is 1- log[5](x) and the right hand side is log[5](x-1). So, since 1 is log[5](5), log[5](5/x)=log[5](x-1). Remove logs: 5/x=x-1. Multiply through by x: 5=x^2-x. Completing the square: x^2-x+1/4=5+1/4, (x-1/2)^2=21/4, x-1/2=sqrt(21)/2 and x=(1/2)(1+sqrt(21)) or (1/2)(1-sqrt(21)). However, the second solution makes x negative, and we can't take logs of a negative number. Therefore the solution is x=1/2(1+sqrt(21))=2.7913.
If the question is 1-ln(5x)=ln(5(x-1)), then ln(e/5x)=ln(5(x-1)), e/5x=5x-5, e=25x^2-25x, 25(x^2-x+1/4)=e+25/4, 5(x-1/2)=sqrt(e+6.25), x-1/2=sqrt(e+6.25)/5, x=1/2+(sqrt(e+6.25))/5=1.0989. For any other base replace e with the base.