Question

If log \(\frac{a}{b}\)+log \(\frac{a}{b}\) log (a+b) , then

• A. a+b=1
• B. a-b=1
• C. a=b
• D. None of these

Answer 1

The Correct Option is A

We are given the equation \[ \log_a \frac{a}{b} + \log_b \frac{b}{a} = \log(a+b). \] Using the property of logarithms \(\log_a \frac{a}{b} = 1 - \log_a b\) and similarly, \(\log_b \frac{b}{a} = 1 - \log_b a\), we can rewrite the given equation as: \[ (1 - \log_a b) + (1 - \log_b a) = \log(a+b). \] Simplifying: \[ 2 - (\log_a b + \log_b a) = \log(a+b). \] Using the change of base formula \(\log_a b = \frac{1}{\log_b a}\), we get: \[ \log_a b + \log_b a = 1. \] Thus, the equation becomes: \[ 2 - 1 = \log(a+b), \] which simplifies to: \[ 1 = \log(a+b). \] Therefore, \(a + b = 10^1 = 1\)

So. the correct option is (A): a+b=1