You have a Taylor series to log( (1+x)/(1-x) ). If you put in x=1/3 you'll see that (1+x)/(1-x) = 2. So you're actually approximating log(2). The m=5 says to evaluate 5 terms. When you add up the terms, you'll get a good guess for log(2). The remainder term tells you how bad your guess could possibly be - that's where you get the bounds on log(2).