Question

The value of \(log_a\bigg(\frac{a}{b}\bigg)+log_b\bigg(\frac{b}{a}\bigg)\), for \(1<a≤b\) cannot be equal to

• A. -0.5
• B. 1
• C. 0
• D. -1

Answer 1

The Correct Option is B

Step 1: Expand and Simplify

Using logarithmic identities:

\[ \log_a\left(\frac{a}{b}\right) = \log_a(a) - \log_a(b) = 1 - \log_a(b) \] \[ \log_b\left(\frac{b}{a}\right) = \log_b(b) - \log_b(a) = 1 - \log_b(a) \]

Step 2: Add Both Expressions

\[ \log_a\left(\frac{a}{b}\right) + \log_b\left(\frac{b}{a}\right) = (1 - \log_a(b)) + (1 - \log_b(a)) \] \[ = 2 - (\log_a(b) + \log_b(a)) \]

Step 3: Let \( \log_a(b) = x \)

Then using change of base:

\[ \log_b(a) = \frac{1}{x} \] \[ \Rightarrow \text{Expression becomes: } 2 - \left(x + \frac{1}{x} \right) \]

Step 4: Analyze the Function

Let: \[ f(x) = x + \frac{1}{x} \] Using AM–GM Inequality: \[ x + \frac{1}{x} \geq 2 \text{ for all } x > 0 \] Therefore: \[ 2 - \left(x + \frac{1}{x} \right) \leq 0 \] So the maximum value of the expression is: \[ 2 - 2 = 0 \]

Final Conclusion

Since the expression is always ≤ 0, it can never be equal to 1.

✅ Final Answer:

\[ \boxed{1 \text{ is not a possible value}} \]