Log linear approximation

Question

I was reading on Likelihood ratio text, and I found an example wit Bernouli trials, along the simplification they did this: Click to view Image

I cant figure out how they did that. I don't want the direct answer, hints are enough.

Answer 1

The first-order (i.e. linear) approximation of a differentiable function $f$ around $1$ is given by

$$f(1+x)\approx f(1)+f'(1)x$$

If $f=\ln$ you get the approximation:

$$\ln(1+x)\approx x.\tag{1}$$ for small $x$.

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For the second case we have

$$\ln\hat{p}-\ln p_0=\ln\frac{\hat{p}}{p_0}=\ln\left(1+\left[\frac{\hat{p}}{p_0}-1\right]\right).$$

Applying the approximation $(1)$ with

$$x=\frac{\hat{p}}{p_0}-1=\frac{\hat{p}-p_0}{p_0}$$

then gives the result you want (where the approximation is good for $\hat{p}\approx p_0$ so that $x\approx 0$).