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Archive for the ‘transcendental number theory’ Category

A student I’m tutoring was working unsuccessfully on the following problem from the 2011 Mandelbrot Competition:

Let a, b be positive integers such that \log_a b = (\log 23)(\log_6 7) + \log_2 3 + \log_6 7. Find the minimum value of ab.

After some tinkering, I concluded that the problem as stated has no solution. I am now almost certain it was printed incorrectly: \log 23 should be replaced by \log_2 3, and then we can solve the problem as follows:

\log_a b + 1 = (\log_2 3 + 1)(\log_6 7 + 1) = \log_2 6 \log_6 42 = \log_2 42.

It follows that \log_a b = \log_2 21. Since a, b are positive integers we must have a \ge 2, and then it follows that the smallest solution occurs when a = 2, b = 21. But what I’d like to discuss, briefly, is the argument showing that the misprinted problem has no solution.

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