# Log simplification question

## Main Question or Discussion Point

This is from a worked example involving using Stirlings approximations.

I have the log equation, where c<<1:

S=K(Nln(N)-cNln(cN)-N(1-c)ln(1-c))

In the next line of the example, this is simplified to

S=-kNcln(c)

I've tried a few ways of getting to this, but have had no success. I've tried making the last term 0, but still don't get the right answer.

Any help would be great :) Thanks

## Answers and Replies

Mute
Homework Helper
First off, I think you're missing a factor of N inside the last logarithm. I.e., your expression (aside from the Boltzmann constant) is

$$N\ln N - cN\ln(cN) - N(1-c)\ln(N(1-c))$$

From there a bunch of stuff should cancel, leaving you with

$$-Nc\ln c - N(1-c)\ln(1-c)$$.

Using the fact that c is small, you can show that the first term is much larger than the other term, hence you can neglect the other term.

Redbelly98
Staff Emeritus
Science Advisor
Homework Helper
It looks like they are using

ln(cN) = ln(c) + ln(N)​

and then neglecting the ln(N) terms relative to -ln(c), i.e. N « 1/c.

UniPhysics90, from the context of what you are reading, does that seem a reasonable assumption?