Alg9.4.33

Find the value of $\frac{1}{\log_{ab}(abc)}+\frac{1}{\log_{bc}(abc)}+\frac{1}{\log_{ca}(abc)}\,$

Splitting the denominators,we get

$\frac{\log ab}{\log abc}+\frac{\log bc}{\log abc}+\frac{\log ca}{\log abc}\,$

$\frac{\log ab+\log bc+\log ca}{\log abc}\,$

$\frac{\log (a^2b^2c^2)}{\log abc}\,$

$\frac{2\log abc}{\log abc}\,$

Hence the value of the given expression is 2