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)?

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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