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Question Number 227471 by hardmath last updated on 02/Feb/26 $$\begin{cases}{\mathrm{log}_{\mathrm{3}} \mathrm{x}=\mathrm{a}}\\{\mathrm{log}_{\mathrm{3}} \mathrm{y}=\mathrm{b}}\end{cases}\:\:\:\:\:\Rightarrow\:\:\:\:\mathrm{log}_{\mathrm{3}} \left(\mathrm{x}^{\mathrm{3}} \centerdot\mathrm{y}^{\mathrm{2}} \right)=? \\ $$ Answered by som(math1967) last updated on 02/Feb/26 $${log}_{\mathrm{3}}…

Explain-how-the-following-conversions-take-place-Give-the-necessary-conditions-a-methane-to-tetra-chloromethane-b-Ethanol-to-acetaldehyde-c-Alchol-to-ethane-

Question Number 227466 by Spillover last updated on 01/Feb/26 $${Explain}\:{how}\:{the}\:{following} \\ $$$${conversions}\:{take}\:{place}.{Give} \\ $$$${the}\:{necessary}\:{conditions} \\ $$$$\left({a}\right){methane}\:{to}\:{tetra}\:{chloromethane} \\ $$$$\left({b}\right){Ethanol}\:{to}\:{acetaldehyde} \\ $$$$\left({c}\right){Alchol}\:{to}\:{ethane} \\ $$ Terms of Service…

Let-f-x-1-q-x-p-q-p-q-Z-gcd-p-q-1-q-gt-0-0-x-R-Q-1-Show-that-f-x-is-continuous-function-when-x-R-Q-2-Show-that-f-x-is-not-a-continuous-function-when-x-Q-3-Prove

Question Number 227438 by Lara2440 last updated on 29/Jan/26 $$\mathrm{Let}\:{f}\left({x}\right)=\begin{cases}{\frac{\mathrm{1}}{{q}}\:\:\:\:\:{x}=\frac{{p}}{{q}}\:,\:{p},{q}\in\mathbb{Z}\:,\:\mathrm{gcd}\left({p},{q}\right)=\mathrm{1}\:,\:{q}>\mathrm{0}}\\{\mathrm{0}\:\:\:\:{x}\in\mathbb{R}\backslash\mathbb{Q}}\end{cases} \\ $$$$\: \\ $$$$\mathrm{1}.\:\mathrm{Show}\:\mathrm{that}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{function}\:\mathrm{when}\:{x}\in\mathbb{R}\backslash\mathbb{Q} \\ $$$$\mathrm{2}.\:\mathrm{Show}\:\mathrm{that}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{continuous}\:\mathrm{function}\:\mathrm{when}\:{x}\in\mathbb{Q} \\ $$$$\mathrm{3}.\:\mathrm{Prove}\:\mathrm{function}\:\mathrm{g}\left({x}\right)\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$$$\mathrm{when}\:\mathrm{g}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{that}\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{only}\:\mathrm{in}\:{x}\in\mathbb{Q} \\ $$$$\mathrm{4}.\:\int_{\:\mathbb{R}} \:{f}\left({x}\right)\mathrm{d}{x} \\ $$…