I read that $\log|x|$ is supposedly a typical example of a BMO function. How do I see that it is in BMO actually?
2026-04-07 07:46:04.1775547964
Why is logarithm in BMO
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$\log|x|$ is funny:
Scaling $x$ adds a constant
No matter for us.
Take a cube of size $1$.
If it's placed near $0$,
Then $\log|x|$ is not big.
On a far-away cube
There's a gradient bound
$\log|x|$ is Lipschitz.