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My aunt asked me the following question from a teenager she is tutoring:

Given that $\alpha \log \beta + \beta \log \alpha = \frac{1}{5}$, and $\alpha \beta = 10$, what is the value of $\alpha + \beta$?

I was able to derive that $\log \alpha + \log \beta = \log 10$. But this doesn't seem to lead me any closer to the solution.

I Like to Code
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  • @MatthewConroy See my edit – I Like to Code Feb 09 '18 at 05:52
  • What is the base of the logarithms used here? – Matthew Conroy Feb 09 '18 at 06:17
  • Are you sure the problem wasn't to find the value of $\alpha \beta$? This pair of non-linear equations seems difficult to solve analytically for $\alpha$ and $\beta$. I couldn't figure out a trick to get a solution $\alpha + \beta$ either. Numerically the result is approximately $\alpha = 11.75$ and $\beta = 0.85$, or vice versa. Thus their sum is $\alpha + \beta = 12.6$. – green orange Feb 13 '18 at 12:51

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