Is it possible to construct a unary real function with countably infinitely many zeros using only finitely many operations from $$ \{+, -, \cdot, /, \exp, \log\} $$ and real constants?
I am asking this question because it seems impossible to produce "infinitely oscillating" functions using only these building blocks.
Precise formulation
Formally, let $S$ be the smallest set satisfying the following conditions.
Each member of $S$ is a function $f \colon D \to \mathbb{R}$ for some $D \subseteq \mathbb{R}$.
The identity function $x \mapsto x$ belongs to $S$.
For any real number $r$, the constant function $x \mapsto r$ belongs to $S$.
If functions $f \colon A \to \mathbb{R}$ and $g \colon B \to \mathbb{R}$ belong to $S$, where $A$ and $B$ are subsets of $\mathbb{R}$, then the following functions also belong to $S$:
$f + g$, $f - g$, and $f \cdot g$, defined over $A \cap B$;
$\frac{f}{g}$, defined over $\{x \in A \cap B \mid g(x) \neq 0\}$;
$\exp \circ f$, defined over $A$;
$\log \circ f$, defined over $A \cap (0, \infty)$, where $\log$ denotes the natural logarithm.
The question asks whether there is a function $f \in S$ such that the set $f^{-1}(\{0\})$ is countably infinite. If the answer is positive, I am also interested in whether the same can be done without using $\log$.
Preliminary observations
Using only polynomials, it is easy to construct a function $x \mapsto \prod_k (x - x_k)$ with any given finite zero set $\{x_1, x_2, \dotsc, x_n\}$; however, this construction cannot be completed in finitely many steps for infinite sets. The zero function has uncountably infinitely many zeros rather than countably infinitely many. (I cannot seem to construct another function with uncountably infinitely many zeros either.)
I tried to show that the answer is negative by attempting to establish various invariants. In particular, I tried to show by structural induction that, for any function $f$ thus constructed, it is always possible to partition the domain of $f$ into finitely many real intervals, such that $f$ satisfies certain "nice" properties on each interval. However, I could not get this argument to work after trying many variations of "nice" involving monotonicity, continuity, and differentiability.
Inspiration
This question is a significantly simplified and weakened version of its inspiration, How to define mod without using integer division or rounding. I hope that insights discovered when solving this question will shed light on that question as well.