An equation is formed using ONLY polynomials (including fractional powers), exponentials and logs. One such equation is shown below; a,b,c,d are real constants. $$ ae^{x+b}-3x^2\left(\ln x^5-c\right)+\left(x^{\frac{6}{d}}-\ln\left(10x-d^8\right)\right)+\frac{ax^{-1}+b}{b^4-c^3}=0\qquad a,b,c,d,x\in\mathbb{R}. $$ Changing the constants $a, b, c, d$ clearly gives rise to a different number of roots and there is obviously an upper limit to how many roots are obtainable in this way.
I am trying to show how the number of roots will have an upper limit (without necessarily finding this limit) but not having any luck. Eg. no matter what $a, b, c, d$ are, the equation above cannot have more than $3$ roots.
I appreciate any pointers, advice and help. Thank you.
Yes, this is true, except when it's trivially false.
To make this question more precise, let $f(x,y_1,\ldots,y_n)$ be an expression formed by composition of exponentials, natural logarithms and rational functions.
Then there is a number $N$ such that for every parameter tuple $(b_1,\ldots,b_n)\in \mathbf R^n$, if the solution set $f(x,b_1,\ldots,b_n)$ does not contain any interval, then it has at most $N$ points.
Note that it can contain an interval, for example if $f=b_1x/(x-1)-b_2x/(x-1)$, then when $b_1=b_2$, $f=0$ everywhere except $x=1$, where it is undefined. (Note that in this case, $f(x,b_1,b_2)$ is zero as an element of the function field; I think that may be the only way we have an interval of solutions, but I don't quite see the argument right now.)
The reason why this is true is that every $f$ of this form is definable in the exponential field $(\mathbf R, +, \cdot, \exp)$ which is known to be o-minimal by a celebrated result of Wilkie. This means that the solution set of $f(x,\bar y)=0$ (in $\mathbf R^{1+n}$) admits an o-minimal cellular decomposition, and the number of cells in this decomposition gives $N$ (this is a weak bound, in fact a much smaller bound is more typical).
For some more background on o-minimality and cellular decomposition, see e.g. these notes by Sergei Starchenko (I'm afraid the details are beyond the scope of an answer here).