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