You can show that ln(1/3) = -ln(3) in a couple of ways:
1. As you noted, 1/3 = 3^(-1). Since ln(a^b) = b ln(a), if you let a = 3 and b = -1 you get ln(3^-1) = -1 ln(3)
2. Recall that ln(a/b) = ln(a) - ln(b). If you let a = 1 and b = 3 you get ln(1/3) = ln(1)-ln(3) = 0-3 = -3
Hope this helps.