Question

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

Answer 1

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

⇒ e^loge^x = e ^-2

[ ∵ Inverse property I . e b^logbx = x]

⇒ 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.