In Chaos and Fractals, New Frontiers of Science, 1st Ed., a function is expanded around a point:
I know how to get the result, but not by using the logarithm approximation. How should I use it here?
Note: The original text actually says $\log(x)=x-x^2/2$, but I assumed it's a typo.
This is just a Taylor second order expansion about the appropriate point:
Let $\phi(t) = t \log t + (1-t) \log(1-t)$.
Then Taylor gives $\phi(t) \approx \phi({1 \over 2}) + \phi'({1 \over 2})(t-{1 \over 2}) + {1 \over 2}\phi''({1 \over 2})(t-{1 \over 2})^2$, or explicitly $\phi(t) \approx - \log 2 + 2 (t-{1 \over 2})^2$ (about $t={1 \over 2}$).
(Note that $\phi'({1 \over 2}) = 0$ since $\phi$ is 'even' about $t={1 \over 2}$.)
Since $f(\alpha) = -{1 \over \log 2}\phi({\alpha_\max - \alpha \over \alpha_\max - \alpha_\min})$, you get $ f(\alpha) \approx 1 -{2 \over \log 2} ({\alpha_0-\alpha \over \alpha_\max - \alpha_\min } )^2$.
Note: The appropriate expansion of $\log$ would be $\log t \approx - \log 2 + 2 (t-{1 \over 2}) -2 (t-{1 \over 2})^2$, however it is far easier to work in terms of $\phi$ above as it 'handles' the necessary cancellations.