Question

If f(xy) = f(x)+f(y), and f(e) = 1, then find the value of $f\left(e^2\right)$

___

Answer 1

We are given f(e) = 1, If we put x and y both equal to e then we can proceed.

Let’s put x = y = e

So, f(e.e) = f(e) + f(e)

$f\left(e^2\right)$ = 2

Alternate method:

We saw that if f(xy) = f(x) + f(y), then f(x) = k lnx or f(x) = 0

But f(e) = 1

$\Rightarrow$ f(x) = k lnx

f(e) = k ln(e) = k = 1(given)

$\Rightarrow$ $f\left(e^2\right)$ = $ln\left(e^2\right)$ = 2