Question

The difference between the logarithms of sum of the squares of two positive numbers A and B and the sum of logarithms of the individual numbers is a constant C. If A = B, then C is

• A. 2
• B. 1.3031
• C. log 2
• D. exp (2)

Answer 1

The Correct Option is C

The correct option is (C): log 2
To solve this problem, we can start with the given condition:
1. Expression for the difference in logarithms:
\[ \log(A^2 + B^2) - (\log A + \log B) = C\]
2. Use properties of logarithms:
  The sum of logarithms can be expressed as:
  \[\log(A) + \log(B) = \log(AB)\]
  Thus, the expression becomes:
  \[\log(A^2 + B^2) - \log(AB) = C\]
  This simplifies to:
  \[\log\left(\frac{A^2 + B^2}{AB}\right) = C\]
3. Substituting \( A = B \):
  If \( A = B \), we can substitute \( A \) for \( B \):
  \[\frac{A^2 + A^2}{A \cdot A} = \frac{2A^2}{A^2} = 2\]
4. Finding the logarithm:
  Therefore, we have:
  \[\log(2) = C\]
So, the value of \( C \) is \( \log(2) \). Thus, the answer is Option C: log2.