Plotting growth functions involving logarithms

Question

How I can plot these functions to compare asymptotic behavior visually?

all of the log here is base $2$.

$1- \log (\log(n^n))$

$2- \log(n!)$

$2- (\log(n))^ {\log(n)}$

We know $1 < 2 < 3$.

Answer 1

I'm assuming you don't mean plotting by hand, in which case you can definitely plot the above functions using online graphing calculators. Try Desmos, Wolfram, or GeoGebra(they're just some of the multiple graphing calculators on the net). I was curious to see what the second function might look like, so you can find the images of the three graphs down below: $$1-ln(ln(x^x))$$

$$2-ln(x!)$$

$$2-(ln(x))^{ln(x)}$$

All images are from Desmos Graphing Calculator. You can try it out on your own. I'll leave the comparison to you.