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
$$0.49915\ldots = \sinh\left((\log 2)^2\right)\approx 1/2$$
Best Answer
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 $1.00052649003$.)