Logarithm(Log) is a function inverse to exponentiation, for example if you have 9=3^x and you want to know x then x=log(3)9=2.

Ln is common denomination for log(e), it is called natural logarithm, for example if A=e^x, then x=lnA.

Not sure what constant you're speaking about. If it is k in your 1st post, then k=2W/S^2. If you want to derive this part of formula or whole formula from scratch (don't have your book, but from what you've posted seems that it is done there by some extent) or at least understand how it is done, then you need to take some course of calculus. You probably don't have proficiency to do it now. But anyway when you have made formula don't see why you need that.

If by constant you mean e, then it is not some poker bankroll related number, but it is fundamental mathematical constant. You can read about it on

Wikipedia.

The formula is really only applicable if you now your "true" winrate and std. By true I mean the numbers that taken from the infinite sample. If your sample is finite (obviously), then more hands you have, bigger the probability that the numbers would be closer to their true values. If for example you have 100 hands, you can calculate your winrate, but the probability that it has nothing to do with your true winrate is higher than probability that it has. 100k hands is just rule of thumb, it is very possible to have 100k and even 500k upswings and downswings due to pure variance.

If you want to be rigorous with your brm you also need to expect higher winrates on micros than on higher limits and adjust your brm accordingly. I believe on NL2 and NL5 you could have winrate 20+bb/100, but that would be hardly possible higher.

I also took mathematical approach to my BRM, but go futher than just using poor RoR formula and derived method to calculate BRM that will give theoretically the highest speed of moving through the limits while preserving desired risk that I would go on the limit lower at the highest point of a given limit. It would take too much time to explain how I had done this and requires advanced math, but this what I'd got and what I'm using:

This is calculated with assumptions of 5bb/100 expected WR on a limit, that std grows with the rate of 5bb/100 by each limit and 5% desired risk of droping to lower limit at the highest point of higher limit. Also I haven't done math for NL 2 and 5 and put numbers that feels reasonable for me. Basically I go on a limit when reach $ in first column and go to lower limit when rich $ in second column.