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Math Help - Distance problem

  1. #1
    Member thaopanda's Avatar
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    Distance problem

    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^*.

    Thanks a bunches!
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  2. #2
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    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|

    You should be able to do the other two properties.
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