Example: The excess Gibbs energy of a binary liquid mixture at $T$ and $P$ is given by:

$\frac{G^E}{R}T=(-2\mathrm{.}6x_1-1\mathrm{.}8x_2)x_1{x}_2$

$($a$)$ Find expressions for $ln{}_1$ and $ln{}_2$ at $T$ and $P.$

$($b$)$ Show that when these expressions are combined in accord with below Eq$.$;

$\frac{G^E}{RT}={\displaystyle \underset{i}{\overset{?}{}}}{x}_{i}ln{}_i$

$,$ the given equation for $G\frac{E}{R}T$ is recovered.