TITLE: Finite number of zeros?
QUESTION [13 upvotes]: Is it true that any nonzero function $f: \mathbb R \to \mathbb R$ which is either:
1) constant, or
2) a polynomial, or
3) $\exp$, or
4) $\log$, or
5) any finite combination of the above using addition, subtraction, multiplication, division and composition, (and individually considered as functions from $\mathbb R$ to $\mathbb R$),
Has a finite number of zeros?

Another attempt. 
I'm actually interested in asymptotic behavior of functions, so a function f(x), as far as I'm concerned, is any expression constructed using the syntax below, such that starting from some point c>0, f(x) is defined for all x>c and takes real values everywhere in this range.
A function has the following syntax:

F --> real number
F --> exp
F --> ln
F --> -F
F --> F + F
F --> F(F)

Hope this specifies exactly what I mean and excludes everything I DON'T.
Any help is appreciated.

REPLY [5 votes]: All the functions under consideration are definable in $\mathbb{R}_{\exp}$, which is o-minimal. Thus their zero sets are finite boolean combinations of intervals. Since the functions are also all analytic, their zero sets are also discrete. Thus the only possible zero sets are finite.