Let f : R⁺ → R, where R⁺ is the set of all positive real numbers, be such that f(x) = log _e x.
Determine
(i) the image set of the domain of f
(ii) {x : f(x) = – 2}
(iii) whether f(xy) = f(x) + f(y) holds.

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\log }_{\mathrm{e}}\mathrm{x} \\ \mathrm{As} \log \mathrm{is} \mathrm{defined} \mathrm{for} \mathrm{positive} \mathrm{value} \mathrm{only} , \mathrm{hence} \\ \mathrm{Domain} \mathrm{is} \mathrm{x}\in \left(0,\infty \right) \\ \mathrm{As} \mathrm{we} \mathrm{know} \mathrm{y}={\log }_{\mathrm{e}}\mathrm{x} \mathrm{gives} \mathrm{x}={\mathrm{e}}^{\mathrm{y}} \\ \mathrm{and} {\mathrm{e}}^{\mathrm{y}} \mathrm{is} \mathrm{positive} \mathrm{for} \mathrm{all} \mathrm{value} \mathrm{of} \mathrm{y}, \mathrm{hence} \mathrm{x}={\mathrm{e}}^{\mathrm{y}} \mathrm{will} \mathrm{be} \mathrm{defined} \mathrm{for} \mathrm{all} \mathrm{y}. \\ \mathrm{Hence} \mathrm{range} \left(\mathrm{image} \mathrm{of} \mathrm{domain} \right) \mathrm{is} \left(-\infty ,\infty \right) \\ \left(\mathrm{i}\right) \\ \mathrm{f}\left(\mathrm{x}\right)=-2 \\ \Rightarrow {\log }_{\mathrm{e}}\mathrm{x}=-2 \\ \Rightarrow \boxed{\mathrm{x}={\mathrm{e}}^{-2}} \\ \left(\mathrm{iii}\right) \\ \mathrm{f}\left(\mathrm{x}\right)= {\log }_{\mathrm{e}}\mathrm{x} \\ \mathrm{f}\left(\mathrm{y}\right)= {\log }_{\mathrm{e}}\mathrm{y} \\ \mathrm{f}\left(\mathrm{xy}\right)= {\log }_{\mathrm{e}}\left(\mathrm{xy}\right) \\ ={\log }_{\mathrm{e}}\mathrm{x}+{\log }_{\mathrm{e}}\mathrm{y} \left[\mathrm{As} {\log }_{\mathrm{a}}\left(\mathrm{mn}\right)={\log }_{\mathrm{a}}\mathrm{m}+{\log }_{\mathrm{a}}\mathrm{n}\right] \\ =\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{y}\right) \\ \mathrm{Hence} \\ \mathrm{f}\left(\mathrm{xy}\right)=\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{y}\right) \mathrm{holds} \end{gathered}$$

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