A material coordinate is denoted by $X$ and a spatial coordinate by $x = \theta(X, t)$.
With any function $f(x,t)$ we can associate a function $F(X, t)$:
$f(x, t) = f(\theta(X, t), t) = F(X, t)$
We have that in spatial coordinates the contininuity, or mass conservation, equation is given by
$\rho_t + (\rho v)_x = 0$
In material coordinates the density will be denoted by $R(X, t) = \rho(\theta(X, t), t)$.
Now I have to show that the continuity equation transforms into
$R_t + R\frac{\partial}{\partial t} \ln(1 + U_X) = 0$
$v_x = \frac{\partial}{\partial t} \ln(1 + U_X)$
But from here I am stuck. Any help by finding the above continuity equation would be grateful. Thanks in advance.
The continuity equation can be expanded $$ \rho_t+\rho_x v+\rho v_x=0\tag1 $$ We have $$ R_t=\rho_t+\rho_x\theta_t=\rho_t+\rho_x v\tag2 $$ while you already proved that $$ v_x=\frac{\partial}{\partial{t}}\ln(1+U_X)\tag3 $$ taking into account $R=\rho$ (with the right variable dependence) and substituting $(2)$ and $(3)$ into $(1)$, we have $$ R_t+R\frac{\partial}{\partial{t}}\ln(1+U_X)\tag3=0 $$