An ambitious and amateur question regarding closed form solution of equations involving finite exponential terms.
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Does this count: $e^{x}=1$ only has solution $x=0$ (or $x=2\pi ik$ if you count complex number). I'm not sure exactly what you're looking for. – calcstudent Jan 02 '20 at 06:08
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Do you mean some $z\in \Bbb{C}$ such that $\sum_{n=1}^N a_n e^{b_n z}=0$ ? If the $a_n,b_n$ are any kind of complex numbers then given $z$ we can always choose $a_N$ such that ... thus you'd need additional constraints to make the problem interesting. – reuns Jan 02 '20 at 06:39
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1see Wikipedia which has several examples. – almagest Jan 02 '20 at 07:22