# Log Approximation

by eep
Tags: approximation
 P: 228 Hi, In his notes, our teacher makes this approximation: $$\log(1 + 3e^{-2\frac{E_o}{\tau}}) \approx \log(3e^{-2\frac{E_o}{\tau}})$$ For $\tau << E_o$ 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...
 Emeritus Sci Advisor PF Gold P: 16,101 For many purposes, $x \approx x+1$ when x is large.
 P: 228 But x isn't large in this case?
Mentor
P: 4,499

## Log Approximation

There doesn't appear to be much. What's the context? Is $$3e^{-2\frac{E_o}{\tau}}$$ very large?

 Emeritus Sci Advisor PF Gold P: 16,101 I thought maybe you had forgotten $\tau < 0$. If the argument to log isn't large, then that's not a good approximation.
 P: 228 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 $\tau << E_o$ 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.
 HW Helper P: 2,566 Are you sure there's a log around that second expression? Because $log(1+x)\approx x$ for x very small.