There is something wrong possibly either with me or with Wikipedia.
Wikipedia's article on the strong three exponentials conjecture
defines $L^\ast$ as the set of all complex numbers of the form
$$\beta_0+\sum_{i=1}^n \beta_i\log\alpha_i$$ where all the $\beta_i$ and $\alpha_i$ are algebraic and every branch of the logarithm is considered.
The strong three exponentials conjecture meanwhile states that if $x_1, x_2$, and $y$ are non-zero complex numbers with $x_1/x_2$ and $y/x_2$ both being transcendental, then at least one of the three numbers $x_1 y, x_2 y, x_2/x_1$ is not in $L^\ast$.
This appears a counterexample:
$$ x_1 =\sqrt{2} ,x_2=\frac12 \sqrt{2}\log{2}, y=1$$
$x_1/x_2$ and $y/x_2$ contain $\log{2}$ and are transcendental. The products are: $$ \begin{aligned} x_1 y =& \sqrt{2} \\ x_2 y =& \frac12 \sqrt{2}\log{2}\\ x_2 / x_1 =& \frac12 \log{2} & \end{aligned} $$
and these are visibly in $L^\ast$.
Is this a counterexample?
Is there a more serious reference for the strong three exponentials conjecture?
Couldn't find it on the web and Wikipedia is not allways reliable source.
