[Math] How to compare logs without a calculator

Question

I have multiple different log sums that I need to evaluate. How would I calculate the following without using a calculator or log tables?

Answer 1

Define the function $f:(0,1) \rightarrow \mathbb{R}$ by $f(x) = -(x \log(x) + (1-x) \log(1-x))$.

Note that $$f(x) = f(1-x)$$ Then$$f'(x) = \log(1-x) - \log(x)$$ And$$f''(x) = -(\frac{1}{x}+\frac{1}{1-x})$$ It follows from this that $f$ hits a maximum at $x=\frac{1}{2}$, is strictly increasing on $x<\frac{1}{2}$.

Your problem is (presumably) to compare $x = f(\frac{2}{5})$, $y = f(\frac{1}{5})$ and $z = f(\frac{2}{5})$. Note that $\frac{1}{5} < \frac{2}{5} < \frac{1}{2}$. Hence $y < x = z$.