Question:
Let f : R+ → R, where R+ is the set of all positive real numbers, such that f(x) = loge x. Determine
(a) the image set of the domain of f
(b) {x : f(x) = −2
(c) whether f(xy) = f(x) : f(y) holds
Solution:
Given:
f : R+ → R
and f (x) = logex .............(i)
(a) f : R+ → R
Thus, the image set of the domain f = R .
(b) {x : f (x) =
⇒ f (x ) =
From equations (i) and (ii), we get :
$\log _{e} x=-2$
$\Rightarrow x=e^{-2}$
Hence, $\{x: f(x)=-2\}=\left\{e^{-2}\right\} . \quad\left[\right.$ Since $\left.\log _{a} b=c \Rightarrow b=a^{c}\right]$
(c) f (xy) = loge(xy) {From(i)}
= logex + logey [Since logemn = loge m + logen]
= f (x) + f (y)
Thus, f (xy) = f (x) + f (y)
Hence, it is clear that f (xy) = f (x) + f (y) holds.