Q64 / 110 · VITEEE 2024 · Mathematics

If $a, b$ and $c$ are distinct positive numbers, not equal to unity. Such that $a b c=1$, then the value of $\log _b a \cdot \log _c a+\log _c b$ $\cdot \log _a b+\log _a c \log _b c$ is

  • A
    0
  • B
    1
  • C
    2
  • Correct answer
    D
    3

Solution

$\begin{aligned} & \text { } \because a b c=1 \\ & \Rightarrow \log a+\log b+\log c=0 \\ & \text { that means, } \\ & \log ^3 a+\log ^3 b+\log ^3 c=3 \log a \cdot \log b \cdot \log c \\ & \Rightarrow \frac{\log ^2 a}{\log b \cdot \log c}+\frac{\log ^2 b}{\log a \cdot \log c}+\frac{\log ^2 c}{\log a \cdot \log b}=3 \\ & \Rightarrow \frac{\log a \cdot \log a}{\log b \cdot \log c}+\frac{\log b \cdot \log b}{\log a \cdot \log c}+\frac{\log c \cdot \log c}{\log a \cdot \log b}=3 \\ & \Rightarrow \quad \frac{\log _b a}{\log _a c}+\frac{\log _c b}{\log _b a}+\frac{\log _a c}{\log _c b}=3 \\ & \Rightarrow \log _b a \cdot \log _c a+\log _c b \cdot \log _a b \\ & +\log _a c \cdot \log _b c=3 \end{aligned}$

Hence, the correct answer is option: 'D'.

More Basic of Mathematics Questions