The Special Mean

Algebra Level 4

Given two positive reals \(a\) and \(b\), the special mean or means are all \(\lambda\) such that \(\log_{a}{\lambda}=\log_{\lambda}{b}\) or \(\log_{b}{\lambda}=\log_{\lambda}{a}.\)

Given that at least one special mean of \(a\) and \(b\) exists, are all special means of \(a\) and \(b\) between \(a\) and \(b\) (inclusive)?

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

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