The Special Mean

Algebra Level 4

Given two positive reals aa and bb, the special mean or means are all λ\lambda such that logaλ=logλb\log_{a}{\lambda}=\log_{\lambda}{b} or logbλ=logλa.\log_{b}{\lambda}=\log_{\lambda}{a}.

Given that at least one special mean of aa and bb exists, are all special means of aa and bb between aa and bb (inclusive)?


Note: This is similar to the way the arithmetic mean of aa and bb is defined as the λ\lambda such that bλ=λa,b-\lambda=\lambda-a, and the geometric mean of aa and bb as the λ\lambda such that bλ=λa.\frac{b}{\lambda}=\frac{\lambda}{a}.

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