  Iterative equation is an equality with an unknown function and its iterates.
There were not found a result on iterative equations with multiplication of
iterates of the unknown function on $\mathbb{R}$. In this paper we use an
exponential function to reduce the equation in conjugation to the well-known
form of polynomial-like iterative equation, but we encountered two
difficulties: the reduction restricts our discussion of the equation to
$\mathbb{R}_+$; the reduced polynomial-like iterative equation is defined on
the whole $\mathbb{R}$ but known results were given on comapct intervals. We
revisit the polynomial-like iterative equation on the whole $\mathbb{R}$ and
give existence, uniqueness, stability and construction of continuous solutions
of our equation on $\mathbb{R}_+$. Then we technically extend our solutions
from $\mathbb{R}_+$ to $\mathbb{R}_-$.
