Is it possible to solve equations of the form: $a^x+b^x+c=0,\;abc\neq0$ with analytical methods; if so, how is this done?
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What are you solving for? – Tim Ratigan Dec 16 '13 at 17:45
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You can't generally solve this analytically, however you can find "easy" solutions for special values of $a$ and $b$. Like $a$ = $b$ implies $a^x = -c/2$. Then using logarithm you are done. – user88595 Dec 16 '13 at 17:47
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@TimRatigan Solving for x. – Natanael Dec 16 '13 at 17:48
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Do these kinds of equations have a name? – Natanael Dec 16 '13 at 17:49
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There are infinitely-many solutions for $x=2$, and, of course, for $x=1$. Just a small data point. – user99680 Dec 16 '13 at 17:52
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If a is a power of b, or vice-versa, then the transcendental equation can be reduced to a polynomial one. Otherwise its solutions cannot even be expressed in terms of the Lambert W function, so your only hope is to solve it by means of numerical algorithms.
Lucian
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Yes, you're right. But even so, we must bear in mind that not all polynomial equations have analytical solutions, i.e., their roots are not expressible in radicals. – Lucian Dec 16 '13 at 18:20