properties about number of solutions of transcendental equations

Question

Consider a transcendental equation $f(x)=c$ , where $c$ is a constant and $f(x)$ is a transcendental function whose the radius of convergence of its taylor series is $\infty$ , once it has solutions, does it should have infinitely many solutions (include complex solutions)?

Answer 1

I think this gives the picture:

If $f$ is an entire function that is not a polynomial then $f$ assumes every complex value, with one exception, an infinite number of times.

(Conway's book on complex analysis)

Answer 2

Not at all: the equation $$e^x=\sum_{n=1}^\infty\frac{x^n}{n!}=c$$ has one unique solution for each and every $\,\,c>0\,$