First slide

Introduction to real valued functions

Question

$If x, y, z are positive real numbers such that \log_{2 x} z = 3, \log_{5 y} z = 6 and \log_{x y} z = 2 / 3 then the value of (1 / 2 z) is$

Moderate

Solution

$\begin{array}{l} z = 8 x^{3}, z = 5^{6} y^{6}, z = x^{2 / 3} y^{2 / 3} \\ ∴ x = \frac{z^{1 / 3}}{2}, y = \frac{z^{1 / 6}}{5}, z = \frac{1}{2^{2 / 3}} z^{2 / 9} \cdot \frac{z^{2 / 18}}{5^{2 / 3}} \\ ∴ z^{2 / 3} = \frac{1}{(10)^{2 / 3}} \\ \Rightarrow z = \frac{1}{10} \end{array}$

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