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Question Number 1180 by 934111 last updated on 11/Jul/15
$${f}\left({x}^{\mathrm{2}} \right)−{f}\left({x}\right)=\mathrm{1} \\ $$
Commented by 123456 last updated on 11/Jul/15
$$\left\{\mathrm{0},\mathrm{1}\right\}\notin\mathrm{D}\left({f}\right) \\ $$
Answered by prakash jain last updated on 13/Jul/15
$${f}\left({x}\right)=\mathrm{log}_{\mathrm{2}} \mathrm{log}_{{k}} {x} \\ $$$${f}\left({x}^{\mathrm{2}} \right)=\mathrm{log}_{\mathrm{2}} \mathrm{log}_{{k}} {x}^{\mathrm{2}} \\ $$$$=\mathrm{log}_{\mathrm{2}} \left(\mathrm{2log}_{{k}} {x}\right)=\mathrm{1}+\mathrm{log}_{\mathrm{2}} \mathrm{log}_{{k}} {x}=\mathrm{1}+{f}\left({x}\right) \\ $$
Commented by Rasheed Ahmad last updated on 13/Jul/15
$${Appreciation}\:{for}\:{your}\:{intelligence} \\ $$$${and}\:{knowledge}.\:{I}\:{would}\:{like}\:{to} \\ $$$${ask}\:{you}\:{how}\:{did}\:{you}\:{determine} \\ $$$${that}?\:{Is}\:{there}\:{any}\:{method}\:{for} \\ $$$${solving}\:\:{such}\:{eqns}? \\ $$
Commented by prakash jain last updated on 13/Jul/15
$$\mathrm{Derived}\:\mathrm{using}\:{f}\left({xy}\right)={f}\left({x}\right)+{f}\left({y}\right) \\ $$$$\Rightarrow{f}\left({x}\right)=\mathrm{log}\:{x} \\ $$$$\mathrm{If}\:\mathrm{multiplication}\:\mathrm{of}\:\mathrm{a}\:\mathrm{variable}\:\mathrm{results}\:\mathrm{in} \\ $$$$\mathrm{addition}\:{f}\left({x}^{\mathrm{2}} \right)={f}\left({x}\right)+\mathrm{1}\:\mathrm{then}\:\mathrm{function}\:\mathrm{will} \\ $$$$\mathrm{be}\:\mathrm{logarithmic}. \\ $$
Commented by Faaiz Soomro last updated on 13/Jul/15
$${Thanks}\:{Parkash}\:{Jain}.{I}\:{am}\:{learning}\:{from}\:{you}. \\ $$