Discussion

Question:

The value of⁡ $\log_{a} (\frac{a}{b}) + \log_{b} (\frac{b}{a})$ , for 1 < a ≤ b cannot be equal to

Explanation:

$\log_{a} (\frac{a}{b}) + \log_{b} (\frac{b}{a})$ = 1 - log_ab + 1 - log_ba = 2 – (log_ab + log_ba)

We know logab is positive and sum of a positive number and its reciprocal is always ≥ 2, i.e.,

∴ log_ab + log_ba ≥ 2

⇒ 2 – (log_ab + log_ba) ≤ 2 – 2

⇒ 2 – (log_ab + log_ba) ≤ 0

∴ 2 – (log_ab + log_ba) cannot take positive value i.e., it cannot be equal to 1 (option (b))

Hence, option (c).

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