The question asks to identify the type of function \(f\) that satisfies the functional equation \(f(xy) = f(x) + f(y)\) for all valid inputs \(x\) and \(y\). This equation is a specific property that only certain types of functions fulfill.

Understanding the Functional Equation f(xy) = f(x) + f(y)

The equation \(f(xy) = f(x) + f(y)\) relates the function's value at the product of two variables to the sum of its values at each individual variable. We need to examine the given types of functions to see which one exhibits this property.

Checking Polynomial Functions for f(xy) = f(x) + f(y)

A general polynomial function is of the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\). Let's consider a simple non-constant example, say \(f(x) = x^2\).

• Left side: \(f(xy) = (xy)^2 = x^2 y^2\).
• Right side: \(f(x) + f(y) = x^2 + y^2\).

Generally, \(x^2 y^2 \neq x^2 + y^2\) (for example, if \(x=2, y=3\), \(f(xy) = (6)^2 = 36\), but \(f(x) + f(y) = 2^2 + 3^2 = 4 + 9 = 13\)). This property does not hold for polynomial functions in general (except for the trivial case \(f(x) = 0\), which is a constant polynomial, but the property usually refers to non-trivial cases or holds for specific domains where other functions are defined).

Checking Trigonometric Functions for f(xy) = f(x) + f(y)

Consider a common trigonometric function like \(f(x) = \sin(x)\).

• Left side: \(f(xy) = \sin(xy)\).
• Right side: \(f(x) + f(y) = \sin(x) + \sin(y)\).

In general, \(\sin(xy) \neq \sin(x) + \sin(y)\) (for example, if \(x = \pi/2, y=1\), \(f(xy) = \sin(\pi/2) = 1\), but \(f(x) + f(y) = \sin(\pi/2) + \sin(1) = 1 + \sin(1)\), which is not 1). Trigonometric functions do not satisfy this functional equation.

Checking Exponential Functions for f(xy) = f(x) + f(y)

Consider an exponential function \(f(x) = a^x\) where \(a > 0\) and \(a \neq 1\).

• Left side: \(f(xy) = a^{xy}\).
• Right side: \(f(x) + f(y) = a^x + a^y\).

In general, \(a^{xy} \neq a^x + a^y\) (for example, if \(a=2, x=2, y=3\), \(f(xy) = 2^{6} = 64\), but \(f(x) + f(y) = 2^2 + 2^3 = 4 + 8 = 12\)). Exponential functions do not satisfy this functional equation.

Logarithmic Functions and the Property f(xy) = f(x) + f(y)

Consider a logarithmic function \(f(x) = \log_b(x)\) where \(b > 0\) and \(b \neq 1\), and typically \(x > 0, y > 0\) for the logarithms to be defined for real numbers.

• Left side: \(f(xy) = \log_b(xy)\).
• Right side: \(f(x) + f(y) = \log_b(x) + \log_b(y)\).

A fundamental property of logarithms states that the logarithm of a product is the sum of the logarithms: \(\log_b(xy) = \log_b(x) + \log_b(y)\). Therefore, the logarithmic function \(f(x) = \log_b(x)\) directly satisfies the functional equation \(f(xy) = f(x) + f(y)\).

Summary of Function Types and the Property

Based on our analysis, the functional equation \(f(xy) = f(x) + f(y)\) is a defining characteristic of logarithmic functions.

Therefore, the functional equation \(f(xy) = f(x) + f(y)\) is true for logarithmic functions.

Function Property Revision Table

Here's a quick review of key properties for different function types:

• Polynomials: Often involve sums of power terms (\(x^n\)). Don't have a simple general property for \(f(xy)\).
• Trigonometric: Relate angles to ratios of sides in triangles. Have addition formulas (e.g., \(\sin(A+B)\)), but not the \(f(xy) = f(x) + f(y)\) form directly for multiplication inside the function.
• Exponential: Have the property \(a^{x+y} = a^x \cdot a^y\), which relates the function of a sum to the product of functions, not the property needed here.
• Logarithmic: Defined as the inverse of exponential functions. The property \(\log_b(xy) = \log_b(x) + \log_b(y)\) is a core rule derived from exponential properties.

Additional Information on Functional Equations

The equation \(f(xy) = f(x) + f(y)\) is a famous functional equation. It is a variant of Cauchy's functional equations. For continuous functions defined on positive real numbers, the solutions are of the form \(f(x) = c \log_b(x)\) for some constant \(c\), or equivalently \(f(x) = C \ln(x)\) for some constant \(C\), where \(\ln\) is the natural logarithm. If the domain includes zero or negative numbers, the definition and properties of the function become more complex.