Why is $\ln 20$ so close to $3$?

Question

\I understand that it can be just a coincidence, but maybe there is a reason?

The closest I could find is using the property that $\log_2 x \approx \ln x + \log_{10}x$ and approximate values of $\log_2 10 \approx 3.3$ and $\log_{10}2 \approx 0.3$. Is there a better explanation?

Answer 1

It's because $e^3\approx 20.086$ is close to $20$, or alternately, because $\sqrt[3]{20}\approx 2.714$ is close to $e$.

Answer 2

Using the software program ries I easily found that $$e^\pi-\pi \approx 19.999099979 \approx 20$$ which I think is more accurate and interesting than $$ e^3 \approx 20.085536923 \approx 20.$$ The approximation with $\pi$ is already known and is asked about in MSE question 724872 (Thanks to a comment by user 'Jam').

Note: Here is output from ries 2.99573227355399099:

Your target value: T = 2.99573227355399                       mrob.com/ries

                      x = 3                      for x = T + 0.00426773  {30}
                   x-pi = 1/-7                   for x = T + 0.00300324  {66}
            sqrt(ln(x)) = pi/3                   for x = T - 0.00169507  {71}
                    e x = 5+pi                   for x = T - 0.000607718 {70}
               log_5(x) = 1-1/pi                 for x = T - 0.000152139 {77}
                    e^x = 4*5                    ('exact' match)         {65}
               (3"/x)^2 = 1/ln(phi)              for x = T - 5.12827e-05 {84}
                 e^x+pi = e^pi                   for x = T - 4.50021e-05 {73}
               1/(pi-x) = phi^4                  for x = T - 3.76537e-05 {87}
              sqrt(e^x) = 2 sqrt(5)              for x = T + 4.44089e-16 {80}
  (Stopping now because best match is within 2.66e-15 of target value.)

Note: You may be interested in the following $$ e^{\pi\sqrt{22}} \approx 2508951.99 $$ $$ e^{\pi\sqrt{37}} \approx 199148674.9999 $$ $$ e^{\pi\sqrt{58}} \approx 24591257751.999999822 $$ which have a satisfying explanation. The biggest of these seems to be $$ e^{\pi\sqrt{163}} \approx 262537412640768743.99999999999925 $$ For details read the Wikipedia Heegner number article.