If p/q = xa – b, q/r = yb – c and r/p = zc – a, then what is the value of (a – b) log x + (b – c) log y + (c – a) log z?

If p/q = xa – b, q/r = yb – c and r/p = zc – a, then what is the value of (a – b) log x + (b – c) log y + (c – a) log z? Correct Answer 0

GIVEN:

p/q = x^{a – b}, q/r = y^{b – c} and r/p = z^{c – a}

FORMULA USED:

log (m/n) = log m – log n

log mⁿ = n log m

CALCULATION:

p/q = x^{a – b}

Taking log on both sides:

log p – log q = (a – b) log x     ---- (1)

Similarly,

log q – log r = (b – c) log y      ---- (2)

log r – log p = (c – a) log z      ---- (3)

Now,

(log p – log q) + (log q – log r) + (log r – log p) = (a – b) log x + (b – c) log y + (c – a) log z

⇒ (a – b) log x + (b – c) log y + (c – a) log z = 0