I am having trouble between two steps of a physics formula for multiplicity.
Currently I have: $$ N\ln(N) - (N-q)\ln(N-q) $$ I am solid on how to get here, but the next steps are what follows:
$$ \text{Assume } N\gg q\gg 1 \text{, and use } [ \ln(1+x)\approx x] \text{ when } x \gg 1,\\ N\ln(N)-(N-q)\left(\ln N-\frac{q}{N} \right),\\ \approx q\ln N. $$
I have been staring at these lines for a few hours now. I cannot figure out how to go from the first formula to the last formula. Any help is much appriciated.
First $\ln(1+x)\approx x$ when $x\color{red}{\ll}1$, not the contrary.
The last formula is obtained factoring$N$ in the last log, and using propertes thereof: \begin{align} N\ln N - (N-q)\ln(N-q)&=N\ln(N) - (N-q)\ln N\Bigl(1-\frac qN\Bigr)\\ &=\not N\not{\ln\not N }- (\not N-q)\ln N - (N-q)\ln\Bigl(1-\frac qN\Bigr)\\ &\approx q\ln N-(N-q)\Bigl(-\frac qN\Bigr)= q\ln N -q+\frac{q^2}N\\ &\approx q\ln N\qquad\text{since }\;q\ll 1\;\text{ and }\;\frac qN\ll 1. \end{align}