Is it possible to solve equation in form $$a^x + bx = c$$ algebraically, where $a$, $b$ and $c$ are given, $a, b, c \in \Bbb{N}$ and $x$ is unknown?
If it's solvable algebraically, how would you solve it?
If it's not solvable algebraically, could calculus be used?
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$\begingroup$See the following theorem, which has a simple derivation by linear substitutions into the Lambert W function, the inverse function for $f(x)=xe^x$:
The equation $p^{ax+b}=cx+d$ where $p>0$ and $c,a\ne0$ yields the solution $$x = -\frac{W(-\frac{a\ln p}{c}\,p^{b-\frac{a d}{c}})}{a\ln p} - \frac{d}{c}.$$
Substituting $p\mapsto a$, $a\mapsto1$, $b\mapsto 0$, $c\mapsto-b$, $d\mapsto c$ from your statement gives the solution
$$x = \frac cb-\frac{W(\frac{\ln a}b\,a^{\frac cb})}{\ln a}.$$
Note that since $f(x)=xe^x$ is not injective, the $W$ function actually has several branches, depending on the values of the parameters.
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