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Re: exponential equation
To prove that there are no more we maybe can use calculus here.
2) For this one there looks like there is one root near - 1 / 2. ( see the last image )
3) There looks like there is one root around - 3 / 5. ( see the first image )
As a general rule when an exponential function begins to overpower a polynomial, the polynomial never recovers. I would say that there are no more roots.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.