Did you mean log5(x-1) - 1 = 1 + log5(x-1) ?
Subtract log5(x-1) from both sides and you get:
-1 = 1
That can't happen, so the equation is wrong / unsolvable.
Did you mean log5(x+1) - 1 = 1 + log5(x-1) ?
Subtract log5(x-1) from both sides:
log5(x+1) - log5(x-1) - 1 = 1
A property of logs is log(a) - log(b) = log(a/b)
log5( (x+1)/(x-1) ) -1 = 1
Add 1 to both sides:
log5( (x+1)/(x-1) ) = 2
Another property of logs is that loga(b) = c means a^c = b
5^2 = (x+1)/(x-1)
25 = (x+1)/(x-1)
25(x-1) = x+1
25x - 25 = x + 1
24x - 25 = 1
24x = 26
x = 26/24
x = 13/12