Let $a,b$ be algebraic numbers.

Assume $(a^a,b^b)$ or $(a^b,b^a)$ are algebraically dependent.

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> Is it true that $\frac{\log{a}}{\log{b}}$ is rational?
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Schnauel's conjecture implies $(\log{a},\log{b},a\log{a},b\log{b})$
or $(\log{a},\log{b},a\log{b},b\log{a})$ are linearly dependent.

**EDIT** Gerry Myerson solved the original question. Let in addition
require $a,b$ to be irrational.