Question

how many half lives have passed if only 0.625g remain of the original 40g?

Answer 1

(1/2)^n = .625
n ln(1/2) = ln(.625)
n = .678

makes sense, since one full half-life leave 1/2, which is a bit less than .625

Answer 2

This answer of 0.678 makes no sense to me. By iteration you know that if the sample starts at 40 and ends up at 0.625 it must have gone through six half-lives. Mathematically it is done this way.
2^n = 40/0.625 = 64
n*log 2 = log 64
0.301n = 1.8062
n = 1.8062/0.301 = 6.0