how do i solve (1/2)**x >= 8

For the problem   (0.5)^x  ≥  8

Let me show you two ways,

First way:  using logs:

Since the variable is an exponent, use logs:

     log(.05^x)  ≥  log(8)

The exponent comes out as a multiplier:

     x · log(0.5)  ≥  log(8)

Divide both sides by log(0.5); since log(0.5) is a negative number, reverse the direction of the inequality:

     x  ≤  log(8) / log(0.5)

     x  ≤  -3

Second way:  no logs:

(1/2)^x  ≥  8

2^(-x)  ≥  2^3

  -x  ≥  3

Dividing by -1 requires that we reverse the direction of the inequality:

   x  ≤  -3