What is the inverse function of $f(x)=x-\log(\log(x))$? If we restrict the domain to e.g. $x\in[2,+\infty[$, the function should have an inverse, but I am unable to compute it.
2026-03-29 17:02:54.1774803774
What is the inverse function of $x-\log(\log(x))$?
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We get a very good approximation with $$ g(x) = x + \left( 1 + \frac{1}{x \log x} \right) \log \log x. $$ Note $g(e) = f(e) = e.$ My calculator says $$ g(f(10)) \approx 10.00061672; \; \; f(g(10)) \approx 10.00062308 $$ $$ g(f(100)) \approx 100.0000241; \; \; f(g(10)) \approx 100.0000234 $$ $$ g(f(1000)) \approx 1000.000000; \; \; f(g(1000)) \approx 1000.000001 $$