Let R^{+} be the set of all positive real numbers. Let f : R^{+}→ R : f(x) = log_{e}x. 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) = log_{e}x

To find: (i) Range of f

Here, f(x) = log_{e}x

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_{e}x …(b)

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

log_{e}x = -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_{e}x]

∴ f(xy) = f(x) + f(y) holds.

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