What complexity have a log(sum) shape?

[tn via Digitalmars-d]

On Tuesday, 8 December 2015 at 15:25:28 UTC, Andrei Alexandrescu 
wrote:
> I'm working on the complexity algebra and of course trying to 
> simplify my life :o). One good simplification would be to get 
> rid of log(polynomial_sum) terms such as:
>
> log(n + m)
> log(n^^3 + n1^^2 + n2)
>
> etc.
>
> Do any of these occur in some important algorithms? I couldn't 
> think of any nor find any on the various complexity cheat 
> sheets around.

It seems that for a complexity such as log(n + m) to make sense, 
it should be possible that both n is more than polynomially 
larger than m and that m is more than polynomially larger than s. 
Since obviously if for instance m < O(n^k), where k is a 
constant, then O(log(n + m)) = O(log n).

I guess that such situations are quite rare.