it is trivial to show that the following holds (just take logs of both sides).
$$x^{\log(y)} = y^{\log(x)}$$
I've forgotten some maths but I believe this is the first time I see this property. Is there a name for it?
The equation $x^{\log(y)} = y^{\log(x)}$: does this result have a widely-used name in the literature
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It's easy to name its cause (commutativity of multiplication, as you know) and its consequence (failure of commutativity of exponentiation), but I know of no name for the result itself, and suspect that it has none. Even if it does, it wouldn't be worth using that name if the readership wouldn't know what it meant. – J.G. Mar 03 '22 at 23:27
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Why does the OP receive downvotes? The question is perfectly written. Also, my (updated) answer answers the OP's question and the downvote remains, that's annoying. I'll remove my answer anyway and put it in the comments. – A. P. Mar 03 '22 at 23:41
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2[re-write] With $x,y\in \mathbb{R}^{*}_{+}$ we have $x^{\ln y}=(e^{\ln x})^{\ln y}=(e^{\ln y})^{\ln x}=y^{\ln x}$. The result it has not name, but it's follows a change of basis and commutative form. – A. P. Mar 03 '22 at 23:42