Is there a reason why $\log \phi \approx (\log 2)^2$?

With $\phi=\frac{\sqrt5+1}2$ the golden ratio, we have $$\log_2 \phi = 0.6942\ldots\\ \,\log_e 2 = 0.6931\ldots$$

Equivalently,

$$2^{\log 2} =1.6168\ldots\\ \quad\;\phi = 1.6180\ldots$$

Is it a simple coincidence or is there a deeper reason for this?

To give an idea of the sort of thing I'm looking for, the identity $\frac{1/\sqrt2}2 = \sinh \frac{\log 2}{2}$ is a very good explanation of the observation that $1/\sqrt2 \approx \log 2$. It also allows estimating the error term by expanding the series for $\sinh$.

Edit: Expanding on infinitylord's remark, this equivalent to the observation

$$0.49915\ldots = \sinh\left((\log 2)^2\right)\approx 1/2$$

• It's depend on the problem, which we need to solve. – Michael Rozenberg Jun 12 '17 at 15:54
• I think that I would to see them a lot closer before I got very interested. – badjohn Jun 12 '17 at 16:01
• @MichaelRozenberg: It's a soft question, but to make it a little bit more formal I could ask: "Is there any algebraic equality that could help derive the bound $|(\log 2)^2 - \log \phi| < 1/1000$ or even $1/500$ without resorting to direct calculation of both terms?" – Milor Jun 12 '17 at 16:03
• As an alternative view, $\log(\phi) - \log(2)^2 = \text{csch}(2)^{-1} - 4 (\coth(3)^{-1})^2$ – infinitylord Jun 12 '17 at 16:16
• It is easy to find a lot of very accurate coincidences, for example : $$\phi=\frac{1+\sqrt{5}}{2}\simeq 1.618033988..$$ $$\cos(\sqrt{2}\:e^{-2})-\cos(\sqrt{2}\:e^\pi) \simeq 1.618033988..$$ $$\sqrt{\cosh(\gamma)+\cos(\gamma)}-\frac{\gamma^3}{\sin(5)}\simeq 1.618033988..$$ $$\cosh\left(G^2\sinh(1)\right)+\cos\left(\frac{\sqrt{\pi}}{\cos(3)}\right)\simeq 1.618033988..$$ $\gamma=$Euler-Masheroni constant $\quad;\quad G=$Catalan's constant. From the paper : fr.scribd.com/doc/14161596/Mathematiques-experimentales , where it is shown how to compute such false equalities. – JJacquelin Jun 12 '17 at 17:33

Take a few well-known constants such as $\gamma,1,\phi,2,e,3,\pi$, a few functions $\log,\exp,\sin,\cos$ and the six basic operations $+,-,\times,\div,a^b,\sqrt[b]a$. Form as many simple expressions as you can by combining them. For example, there are more than $6000$ expressions of the form $f(x)\text{ op }g(y)$ many of which fall in a small range, say $[0.1,10]$. Hence you can expect a few pseudo-coincidences.

Below a plot of the values of $2717$ such expressions with a value in the above range. This exhaustive search reveals the interesting

$$\frac{e^\gamma}{e^e}\approx\frac{e}{e^\pi}$$ $$0.117529\approx0.117467$$

which is nothing but $$\gamma-e+\pi\approx 1.$$ (Actually $1.00052649003$.)

• pretty neat! how did you draw that? – Glougloubarbaki Jul 31 '17 at 19:18
• @Glougloubarbaki Excel. – Yves Daoust Jul 31 '17 at 20:26
• +1 I've made some minor edits for brevity. Can you give a few more examples from your search? – Tito Piezas III Aug 1 '17 at 12:49
• @TitoPiezasIII: brevity ? Who cares ? – Yves Daoust Aug 1 '17 at 12:58