Correct Answer - Option 4 : log x

**Concept:**

The arithmetic mean is the sum of all the numbers in a data set divided by the quantity of numbers in that set.

**Arithmetic mean** of two positive numbers a and b is \(\rm \frac{{a\; + \;b}}{2}\)

Logarithmic formula

log mn = log m + log n

\( \rm \log (\frac{m}{n}) = \log m - \ log n\)

**Calculation:**

Given: f(x) = log x

To Find: AM of f(xy) and f(x/y)

f(xy) = log (xy) = log x + log y (∵ log mn = log m + log n)

\( \rm f(\frac{x}{y}) = \log (\frac{x}{y}) = \log x - \ log y\)

Now, AM of f(xy) and f(x/y) = \(\rm \frac{f(xy)+f(\frac{x}{y})}{2}\)

\(= \rm \frac{\log x+ \log y + \log x - \log y}{2}\\ = \frac{2\log x}{2} \\= \log x\)