Question

If x, y, z are distinct positive numbers different from 1, such that (log_y x. log_z x – log_x x) + (log_x y.log_z y – log_y y) + (log_x z. log_y z – log_z z) = 0 then xyz equals 

(a) 100 

(b) –1 

(c) 10 

(d) 1

Answer 1

(d) 1

log_y x. log_z x – log_x x = \(\frac{\text{log}\,x}{\text{log}\,y}\) . \(\frac{\text{log}\,x}{\text{log}\,z}\) - 1 = \(\frac{\text{(log}\,x^2)}{\text{log}\,y.\,\text{log}\,z}\) - 1

Similarly, log_x y.log_z y – log_y y = \(\frac{\text{(log}\,y^2)}{\text{log}\,x.\,\text{log}\,z}\) - 1 and

log_x z. log_y z – log_z z = \(\frac{\text{(log}\,z^2)}{\text{log}\,x.\,\text{log}\,y}\) - 1

∴ LHS = \(\frac{\text{(log}\,x^2)}{\text{log}\,y.\,\text{log}\,z}\) - 1 + \(\frac{\text{(log}\,y^2)}{\text{log}\,x.\,\text{log}\,z}\) - 1 + \(\frac{\text{(log}\,z^2)}{\text{log}\,x.\,\text{log}\,y}\) - 1 

= \(\frac{(\text{log}\,x)^3+(\text{log}\,y)^3+(\text{log}\,z)^3-3\text{log}\,x.\text{log}\,y.\text{log}\,z}{\text{log}\,x.\text{log}\,y.\text{log}\,z}\) = 0          (given)

⇒ (log x)³ + (log y)³ + (log z)³ – 3 log x. log y. log z = 0 

⇒ log x + log y + log z = 0           (if a + b + c = 0, then a³ + b³ + c³ = 3abc) 

⇒ log xyz = 0 ⇒ xyz = 1.