How prove that: $\log_3{5} + \log_2{5}$ is irrational?
$$\log _3{5} + \log _2{5}=\frac{1}{\log _5{3}}+ \frac{1}{\log _5{2}}= \frac{\log _5\left( 3\cdot 2\right) }{\log _53 \cdot \log _52}$$
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2See also http://mathoverflow.net/questions/185540/ – user84413 Dec 23 '15 at 18:22