Suppose we have this:
Greater Number > Lesser Number
Then we also have this:
log(Greater Number) > log(Lesser Number)
Taking the log of each side will not switch the inequality sign. Log(greater number) will still be greater than log(lesser number). What logging each side will do is cut down the value difference between the lesser side and greater side. Example: 1000>>0 but 6.9078 = log(1000) > log(10) =2.3026.
Note that >> means "much much greater than" as opposed to simply "greater than"