logs value of x

log₃(x) + log₃(x-2)= 1         (note   1 =  log₃ [3] )

log₃ [ (x) (x-2) ]  =  log₃ [3]     

Since we have the same logs on both sides, we can forget the logs and solve this :

(x) (x - 2)  = 3     simplify

x^2 - 2x - 3  = 0     factor

(x - 3) ( x + 1)  = 0      and setting each factor to 0, we have that x = 3 or x = -1

Reject -1 since it produces the log of a negative number in the original equation

So, x = 3 is the only solution