  The theory of valued difference fields $(K, \sigma, v)$ depends on how the
valuation $v$ interacts with the automorphism $\sigma$. Two special cases have
already been worked out - the isometric case, where $v(\sigma(x)) = v(x)$ for
all $x\in K$, has been worked out by Luc Belair, Angus Macintyre and Thomas
Scanlon; and the contractive case, where $v(\sigma(x)) > n\cdot v(x)$ for all
$n\in\mathbb{N}$ and $x\in K^\times$ with $v(x) > 0$, has been worked out by
Salih Azgin. In this paper we deal with a more general version, called the
multiplicative case, where $v(\sigma(x)) = \rho\cdot v(x)$, where $\rho (> 0)$
is interpreted as an element of a real-closed field. We give an axiomatization
and prove a relative quantifier elimination theorem for such a theory.
