f(xy) = f(x) + f(y) is true for all

Options

A. Polynomial function
B. Trgonometric functions
C. Exponential functions
D. Logarithmic functions

Answer

To solve the problem where \( f(xy) = f(x) + f(y) \) is true for all \( x \) and \( y \), we need to determine which type of function satisfies this property. We will explore the logarithmic function as a potential solution.

### Step-by-Step Solution:

1. **Understanding the Functional Equation**:
We are given the functional equation \( f(xy) = f(x) + f(y) \). This means that the value of the function at the product of two numbers is equal to the sum of the values of the function at those two numbers.

2. **Assuming a Form for \( f(x) \)**:
Let's assume that \( f(x) \) is of the form \( f(x) = \log(x) \). This is a common function to test for this type of equation.

3. **Calculating \( f(xy) \)**:
If \( f(x) = \log(x) \), then:
\[ f(xy) = \log(xy) \]

4. **Using Logarithmic Properties**:
We can use the property of logarithms that states \( \log(ab) = \log(a) + \log(b) \). Therefore:
\[ f(xy) = \log(xy) = \log(x) + \log(y) = f(x) + f(y) \]

5. **Conclusion**:
Since we have shown that \( f(xy) = f(x) + f(y) \) holds true when \( f(x) = \log(x) \), we conclude that the logarithmic function satisfies the given functional equation.

6. **Identifying the Correct Option**:
Based on our findings, the function that satisfies the equation \( f(xy) = f(x) + f(y) \) for all \( x \) and \( y \) is the logarithmic function. Thus, the correct answer is option D: the logarithmic function.