Question Number 223512 by fantastic last updated on 27/Jul/25
$${If}\:\frac{\mathrm{log}\:{x}}{{y}−{z}}=\frac{\mathrm{log}\:{y}}{{z}−{x}}=\frac{\mathrm{log}\:{z}}{{x}−{y}} \\ $$$${prove}\:{xyz}=\mathrm{1} \\ $$
Answered by mr W last updated on 27/Jul/25
$$\frac{\mathrm{log}\:{x}}{{y}−{z}}=\frac{\mathrm{log}\:{y}}{{z}−{x}}=\frac{\mathrm{log}\:{z}}{{x}−{y}}=\frac{\mathrm{1}}{{k}}\:\:\left({k}\neq\mathrm{0}\right) \\ $$$${y}−{z}={k}\:\mathrm{log}\:{x} \\ $$$${z}−{x}={k}\:\mathrm{log}\:{y} \\ $$$${x}−{y}={k}\:\mathrm{log}\:{z} \\ $$$$\Sigma: \\ $$$$\Rightarrow\mathrm{0}={k}\:\left(\mathrm{log}\:{x}+\mathrm{log}\:{y}+\mathrm{log}\:{z}\right)={k}\:\mathrm{log}\:{xyz} \\ $$$$\Rightarrow\mathrm{log}\:{xyx}=\mathrm{0}\:\Rightarrow{xyz}=\mathrm{1}\:\checkmark \\ $$
Commented by fantastic last updated on 27/Jul/25
$${thanks} \\ $$