Starting equation: 32x - 1 = 1
9
Re-write the right side as 3-2, to get:

32x - 1 = 3-2

If the powers are equal, and the bases (here, 3) are equal the the exponents must also be equal:

2x - 1 = -2
2x = -2+1
2x = -1
x = -½
See Rules for exponents?