Exponential and Logarithmic Functions - Laws of Logarithms

Use real logarithms. In part (a), \(b>0\). In part (b), \(x,y,z>0\), each differs from 1, and \(xyz\ne1\), so every logarithm and denominator is defined.

1. Show that for any \(a>0\) and \(a\ne 1\), it holds that \({{\log }_{{{a}^{r}}}}b=\frac{1}{r}{{\log }_{a}}b\), where \(r\ne 0\).
2. Hence, or otherwise, find the value of \(\frac{4}{{{\log }_{\sqrt{x}}}\left( xyz \right)}+\frac{6}{{{\log }_{\sqrt[3]{y}}}\left( xyz \right)}+\frac{8}{{{\log }_{\sqrt[4]{z}}}\left( xyz \right)}\).