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.