Are all real continous elementary functions included in $$ e^{kx} $$ $k$ is a complex number , $x$ is real variable . This question came to my mind when solving linear higher order ODE. We use a substitution like this to solve those equations and i often hear people say we assume the solution is a exponetial function, although we often get $ \sin, \cos $ functions ( which we get by the magic of euler formula).
If $k$ is a real number we have only the exponential function. If $k$ is equal to $i$ we get $ \sin, \cos $ can we get other all other functions such as $x^n$ or $\log x$ if $k$ is a complex number $a+ib$, then we would have $$ e ^{(a+ib)x}=e^{ax}(\cos bx+i \sin bx) $$
So can we find $a,b$ such this expression is equal to e.g. $ \sqrt x $ for all $x$ . I think it is not possible but solving that equation for $a,b$ seems only possible numerically
$$x=e^{\log{(x)}}$$ Now $\log{x}$ cannot be written as or even approximated by $kx$ for a constant $k$, so $x$ cannot be written in terms of $e^{kx}$.