Do I need to flip the inequality sign when I take log on both sides, which causes both sides to become negative?

Answer 1

log is 10^x; (log 100)=2. If you have 10^(negative)=; 1 would be your numerator. 10^(-2)=1/100 or .01.

Answer 2

No, it is a monotonic transformation. a simple example will help.

1/8< 1/4 = 1/8 = 1/((2^3) = 2^(-3) 1/4 = 2^(-2)

ln(2^-3) = -3ln(2) < ln(2^(-2) = -2ln2

or - 3ln(2)<-2ln(2)