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Logarithmic differentiation is used in mathematics when dealing with infinite products in complex analysis. Note the construction of $f’/f$ from $f$ does not require logarithms of $f$ as a middle step (except intuitively), so it is perfectly fine to use even when $f$ takes non-positive values. The prime number theorem and the link between prime numbers and ...


For the expression $x^x$ we could focus on finding occurrences of $x\ln(x)$. One direction is Stirling's approximation $\ln(N!)\sim N\ln(N)$ so $N!$ is like $N^N$. Another direction is that the prime number theorem gives an estimate for the $n$-th prime $p_n$ as $p_n=n\ln(n)$. Yet another direction involves entropy. Entropy might be given by an expression ...


$n^n$ shows up in combinatorics (number of lists of numbers from 1-n of length n), but I doubt it will have applications for the physical world: $x$ cannot have units, since it doesn't make sense to raise something to a power with units.

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