Hi, In his notes, our teacher makes this approximation: [tex] \log(1 + 3e^{-2\frac{E_o}{\tau}}) \approx \log(3e^{-2\frac{E_o}{\tau}}) [/tex] For [itex]\tau << E_o[/itex] Also, and I don't think this matters, the logs are assumed to be natural logs. I was wondering what the justification for this was...
There doesn't appear to be much. What's the context? Is [tex]3e^{-2\frac{E_o}{\tau}}[/tex] very large? EDIT: In that case, ask your teacher
I thought maybe you had forgotten [itex]\tau < 0[/itex]. If the argument to log isn't large, then that's not a good approximation.
Sorry I hadn't quite finished editing my post when people started replying. We're trying to calculate the partition function for rotational degrees of freedom for a single molecule. So we have an infinite sum which we keep only the first two terms in the [itex]\tau << E_o[/itex] limit (the terms in the log). We then want to calculate the average energy which is where the log comes from, and he then makes that approximation. I guess I'll just have to ask him.
Are you sure there's a log around that second expression? Because [itex]log(1+x)\approx x [/itex] for x very small.