Let $R^*$ := {x $\in$ R : x > 0} and consider d : $R^* \times R^* \rightarrow$ R given by d(x,y) := | $\ln \frac{x}{y}$ |. Show that d is a distance on $R^*$.
2. Note this: $\left| {\ln \left( {\frac{x}{y}} \right)} \right| = \left| {\ln (x) - \ln (y)} \right|$.
$d(x,z) = \left| {\ln (x) - \ln (z)} \right| \leqslant \left| {\ln (x) - \ln (y)} \right| + \left| {\ln (y) - \ln (z)} \right|$