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

Answer: (C)

log 2

Answer 2

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.