Is it possible to write a finite equation consisting ONLY of exponentials, logs and $x^n$ where $n$ is any real number which has an infinite number of non-equal real roots ? (trivial examples like $x-x=0$, $(\sqrt(x)^2)$/$x =1$, imaginary numbers, piecewise functions to be excluded). For instance can an equation like the one below (or a bigger one) have an inf number of real roots? I think not but how does one go about proving it ?
$\frac{exp(x^3-4x).ln(\sqrt(x^x-cosh(2x+1))+1}{exp(2x+1)}$ + $\frac{6.exp(x^3-4x).ln(\sqrt(4x^x-asinh(2-6x))+1}{x^3-exp(9x+1)}$ + $\frac{exp(x^5-4x.ln(\sqrt(8x^x-cosh(24x-16))-9}{564exp(2x-91)}$ - $\frac{2x^6-13x^2+xln(\sqrt(x^2-atanh(29x-1))-1}{x^7-2x^3+9x^6-2x+ln(1+x+x^8)}$ = $(ln(x)^3-1/x)^x$
I am interested in hearing of a general argument and not one specifically for the equation above.
ps: only the functions I have mentioned in line 1 are allowed, all other functions, operators (exc + - / x), integrals, derivatives are disallowed.
The first order theory $\Bbb{R}_{\mathsf{exp}}$ over the language that adds the exponential function $\mathsf{exp}(\cdot)$ to the usual field operators on the reals is know to be o-minimal, which implies that the set of roots of an equation of the sort you are interested in is a finite union of points and intervals (so if there are infinitely many distinct roots they are all contained in a finite set of intervals each of whose points is a root). See the Wikipedia article on exponential fields for references.