Given that f: R⁺→ R such that f(x) = log_ex

To find: (i) Range of f

Here, f(x) = log_ex

We know that the range of a function is the set of images of elements in the domain.

∴ the image set of the domain of f = R

Hence, the range of f is the set of all real numbers.

To find: (ii) {x : x ϵ R⁺ and f(x) = -2}

We have, f(x) = -2 …(a)

and f(x) = log_ex …(b)

From eq. (a) and (b), we get

log_ex = -2

Taking exponential both the sides, we get

⇒ x = e^-2

∴{x : x ϵ R⁺ and f(x) = -2} = {e^-2}

To find: (iii) f(xy) = f(x) + f(y) for all x, y ϵ R

We have,

f(xy) = log_e(xy)

= log_e(x) + log_e(y)

[Product Rule for Logarithms]

= f(x) + f(y) [∵f(x) = log_ex]

∴ f(xy) = f(x) + f(y) holds.