Question

Consider a function f satisfying f(x * y) = f(x) + f(y) where x, y are positive integers, then what is the value of \( \frac{f(3) + f(12){f(12) - f(4) + f(2)} \)?}

• A. 2
• B. 6
• C. 3
• D. 8
• E. 4

Hint:

Recognize that \(f(x \cdot y) = f(x) + f(y)\) is the property of logarithms. You can think of \(f(n) = k \log(n)\). This helps in quickly breaking down composite numbers like \(f(12)\) into sums of their factors, like \(f(3) + f(4)\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The given functional property is \(f(x \cdot y) = f(x) + f(y)\). This is a characteristic property of logarithmic functions. We can use this property to break down the terms in the given expression into simpler components.
Step 2: Detailed Explanation:
We need to simplify the expression: \[ \frac{f(3) + f(12)}{f(12) - f(4) + f(2)} \] Let's first break down the terms \(f(12)\) and \(f(4)\) using the given property.
For \(f(12)\), we can write \(12\) as a product of smaller integers, for example, \(12 = 3 \times 4\).
\[ f(12) = f(3 \times 4) = f(3) + f(4) \] For \(f(4)\), we can write \(4\) as \(2 \times 2\).
\[ f(4) = f(2 \times 2) = f(2) + f(2) = 2f(2) \] Now, substitute these expanded forms back into the original expression.
Numerator: \[ f(3) + f(12) = f(3) + (f(3) + f(4)) = 2f(3) + f(4) \] Substituting \(f(4) = 2f(2)\), we get: \[ 2f(3) + 2f(2) = 2(f(3) + f(2)) \] Denominator: \[ f(12) - f(4) + f(2) \] Substitute \(f(12) = f(3) + f(4)\): \[ (f(3) + f(4)) - f(4) + f(2) = f(3) + f(2) \] Step 3: Final Answer:
Now, we form the fraction with the simplified numerator and denominator.
\[ \frac{2(f(3) + f(2))}{f(3) + f(2)} \] Assuming \(f(3) + f(2) \neq 0\), we can cancel this term from the numerator and the denominator.
\[ = 2 \] Thus, the value of the expression is 2.