# Tagged Questions

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### Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
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### Why does $\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$

I read on Wikipedia that $$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$ to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical ...
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### Big-$\mathcal{O}$ bounding of sums of logarithmic functions

I am reading a text which states that $$\sum \limits_{n \leq X} \left(\log X - \log n \right) = \mathcal{O}(X)$$ I can't quite see why this is true, though I can certainly believe it. Could anyone ...
### Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer?
I read that $$10\frac{\exp(\pi)-\log 3}{\log 2} =318.000000033252\dots \approx 318$$ Is this simply a coincidence or can this somehow be explained?