If (log a / b - c) = (log b / c - a) = (log c / a - b), then a^a b^b c^c equals :

Question

If \(\frac{log\,a}{b-c}=\frac{log\,b}{c-a}=\frac{log\,c}{a-b}\) , then \(a^ab^bc^c\) equals :

(a) –1 

(b) 0 

(c) abc 

(d) 1

Answer 1

(d) 1

Let \(\frac{log\,a}{b-c}=\frac{log\,b}{c-a}=\frac{log\,c}{a-b}\) = k

⇒ log a = k(b – c), log b = k(c – a), log c = k(a – b) 

Now let \(a^ab^bc^c\) = p. Then, 

log p = loga^a + logb^b + logc^c = a log a + b log b + c log c 

= a x k (b – c) + b x k(c – a) + c x k (a – b) 

= k(ab – ac + bc – ba + ca – cb) = 0 

⇒ log p = log 1                   (Putting log 1 for 0) 

⇒ p = 1 ⇒ \(a^ab^bc^c\) = 1.