# 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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