So we have Liouville's Constant:
$L_b=\displaystyle\sum_{n\in\mathbb{Z}^+}b^{-n!}=\left(0.\textbf{1}\textbf{1}000\textbf{1}00000000000000000\textbf{1}00000000...\right)_b$
And let $M$ be the constant defined as: (is there a name for this constant?)
$M_b=\displaystyle\sum_{n\in\mathbb{Z}_{\geq 0}}b^{-\frac{n^2}{2}-\frac{n}{2}-1}=\frac{\theta_2\left(\frac{1}{\sqrt{b}}\right)}{2b^{7/8}}=\left(0.\textbf{1}\textbf{1}0\textbf{1}00\textbf{1}000\textbf{1}0000\textbf{1}00000\textbf{1}000000\textbf{1}...\right)_b$
My question is, are either of $e^{L_b}$ or $e^{M_b}$ transcendental? How would one go about proving or disproving that $e^{L_b}$ or $e^{M_b}$ are transcendental?
Or, if $L$ is any Liouville Number (therefore $L$ is transcendental), is $e^L$ transcendental?
How about $10^L$?