Question

If a, b, c are distinct positive numbers, each different from 1, such that $\lbrack\log_{b}a\log_{c}a - \log_{a}a\rbrack + \lbrack\log_{a}b\log_{c}b - \log_{b}b\rbrack$

$+ \lbrack\log_{a}c\log_{b}c - \log_{c}c\rbrack = 0,$ then abc =

• A. 1
• B. 2
• C. 3
• D. None of these

Answer 1

Answer: (A)

1

Answer 2

$\lbrack\log_{b}a.\log_{c}a - \log_{a}a\rbrack + \lbrack\log_{a}b.\log_{c}b - \log_{b}b\rbrack + \lbrack\log_{a}c\log_{b}c - \log_{c}c\rbrack = 0$

⇒ $x = \frac{9}{4}$

⇒ $\frac{1}{\ln a.\ln b.\ln c}\lbrack(\ln a)^{3} + (\ln b)^{3} + (\ln c)^{3} - 3\ln a.\ln b.\ln c\rbrack = 0$

⇒ $(\ln a)^{3} + (\ln b)^{3} + (\ln c)^{3} - 3\ln a.\ln b.\ln c = 0$

⇒ $\ln a + \ln b + \ln c = 0$

⇒ $\ln(abc)$ = ln 1, $\lbrack a^{3} + b^{3} + c^{3} - 3abc = 0$

⇒ $a + b + c = 0\rbrack$, $\therefore abc = 1$.