What is f(16) equal to?

Calculation:

Given,

The function satisfies the equation \( \frac{f(x)}{f(y)} = \frac{x}{y} \) for all positive real values of x and y, and \( f(2) = 3 \).

We are tasked with finding \( f(16) \).

Using the functional equation, for \( f(4) \), we have:

\( \frac{f(4)}{f(2)} = f\left( \frac{4}{2} \right) = f(2) \)

Since \( f(2) = 3 \), we can calculate \( f(4) \)

\( \frac{f(4)}{3} = 3 \quad \Rightarrow \quad f(4) = 9 \)

Next, to find \( f(16) \), we use the functional equation again:

\( \frac{f(16)}{f(4)} = f\left( \frac{16}{4} \right) = f(4) \)

Since \( f(4) = 9 \), we can calculate \( f(16) \)

\( \frac{f(16)}{9} = 9 \quad \Rightarrow \quad f(16) = 81 \)

Hence, the correct answer is Option 4.