First slide

Types of real valued functions XI

Question

If $\frac{\log x}{b - c} = \frac{\log y}{c - a} = \frac{\log z}{a - b},$ then which of the following is/are true?

Moderate

A

xyz=1
B

$x^{a} y^{b} z^{c} = 1$
C

$x^{b + c} y^{c + a} z^{a + b} = 1$
D

$xyz = x^{a} y^{b} z^{c}$

Solution

Let $\frac{\log_{k} x}{b - c} = \frac{\log_{k} y}{c - a} = \frac{\log_{k} z}{a - b} = p$
$\begin{array}{l} \Rightarrow x = k^{p (b - c)}, y = k^{p (c - a)}, z = k^{p (a - b)} \\ \Rightarrow xyz = k^{p (b - c)} k^{p (c - a)} k^{p (a - b)} \\ = k^{p (b - c) + p (c - a) + p (a - b)} = k^{0} = 1 \\ x^{a} y^{b} z^{c} = k^{pa (b - c)} k^{pb (c - a)} k^{pc (a - b)} = k^{0} = 1 \\ x^{b + c} y^{c + a} z^{a + b} = k^{p (b + c) (b - c)} k^{p (c + a) (c - a)} k^{p (a + b) (a - b)} \\ = k^{0} = 1 \end{array}$

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