Let R+ be the set of all positive real numbers. Let f : R+ R : f(x) = logex. Find

(i) range (f)

(ii) {x : x ϵ R+ and f(x) = -2}.

(iii) Find out whether f(x + y) = f(x). f(y) for all x, y ϵ R.

Given that f: R+ R such that f(x) = logex

To find: (i) Range of f

Here, f(x) = logex

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) = logex …(b)

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

logex = -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) = loge(xy)

= loge(x) + loge(y)

[Product Rule for Logarithms]

= f(x) + f(y) [f(x) = logex]

f(xy) = f(x) + f(y) holds.