Let $\displaystyle R^*$ := {x $\displaystyle \in$ R : x > 0} and consider d : $\displaystyle R^* \times R^* \rightarrow $ R given by d(x,y) := | $\displaystyle \ln \frac{x}{y}$ |. Show that d is a distance on $\displaystyle R^*$.
Thanks a bunches!
Let $\displaystyle R^*$ := {x $\displaystyle \in$ R : x > 0} and consider d : $\displaystyle R^* \times R^* \rightarrow $ R given by d(x,y) := | $\displaystyle \ln \frac{x}{y}$ |. Show that d is a distance on $\displaystyle R^*$.
Thanks a bunches!
Note this: $\displaystyle \left| {\ln \left( {\frac{x}{y}} \right)} \right| = \left| {\ln (x) - \ln (y)} \right|$.
$\displaystyle d(x,z) = \left| {\ln (x) - \ln (z)} \right| \leqslant \left| {\ln (x) - \ln (y)} \right| + \left| {\ln (y) - \ln (z)} \right|$
You should be able to do the other two properties.