It comes up a lot in analytic number theory to manipulate functions with lots of logarithms. In general these look something like
x^e0 * log(x)^e1 * log(log x)^e2 * ...
or sums of such products. For context, these are a special case of Hardy's logarithmico-exponential functions (see Orders of Infinity), which I believe are further generalized by Hardy fields.
The kinds of things I do with these objects is simple: multiplication, addition, composition. It's basically arithmetic, but I'd like to be able to do it automatically for the same reason I like to do arithmetic on the computer: it's faster and less error-prone. For each page of symbols I move around, there's a nontrivial chance I'll make a mistake or misread a figure, and then the whole calculation is lost.
So I was wondering if there was a good way to do this in PARI/GP? I've been able to do some amount of this, poorly, in Mathematica; it doesn't seem to have a built-in notion of what to do, but I can prod it to do what's needed by taking various limits. But at the end of the day none of this is very complicated, so I was hoping to have a better approach.
As an example of what I'm trying to do, take this problem:
The n-th prime, for n >= 39017, is between f_1(n) and f_k(n) with k = 0.9484, with f_v(x) = n*(log n + log log n - k). What is f_v(f_v(n))? By hand, I get n log(n)^2 + 3n log n log log n + ....