Question Number 199956 by cortano12 last updated on 11/Nov/23 Answered by Frix last updated on 11/Nov/23 $$\left(\mathrm{1}\right)\:\Rightarrow\:{x}={y}\left({y}^{\mathrm{2}} +{y}+\mathrm{2}\right)\wedge{y}>\mathrm{0} \\ $$$$\left(\mathrm{2}\right)\:\Rightarrow\:\frac{{x}}{{y}}=\mathrm{4}\vee\frac{{x}}{{y}}=\mathrm{8} \\ $$$$\Rightarrow \\ $$$${x}=\mathrm{4}\wedge{y}=\mathrm{1}\:\vee\:{x}=\mathrm{16}\wedge{y}=\mathrm{2} \\…
Question Number 199801 by Calculusboy last updated on 09/Nov/23 Answered by Frix last updated on 09/Nov/23 $${x}={a}\sqrt{{b}−{a}\sqrt{{b}−{a}\sqrt{{b}−…}}} \\ $$$${x}={a}\sqrt{{b}−{x}} \\ $$$${x}^{\mathrm{2}} ={a}^{\mathrm{2}} \left({b}−{x}\right) \\ $$$${x}^{\mathrm{2}}…
Question Number 198222 by cortano12 last updated on 14/Oct/23 $$\:\:\:\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{5}^{\mathrm{x}} −\mathrm{3}^{\mathrm{x}} \right)\:=\:\mathrm{log}\:_{\mathrm{5}} \left(\mathrm{4}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}\:} \right) \\ $$ Commented by Rasheed.Sindhi last updated on 14/Oct/23…
Question Number 198124 by stevoh last updated on 10/Oct/23 $${solve}\:{for}\:{x}\:{log}\mathrm{100}+{log}\left(\mathrm{2}+{x}\right)=\mathrm{10} \\ $$ Answered by a.lgnaoui last updated on 10/Oct/23 $$\:\boldsymbol{\mathrm{xlog}}\mathrm{100}+\boldsymbol{\mathrm{log}}\left(\mathrm{2}+\boldsymbol{\mathrm{x}}\right)=\mathrm{10} \\ $$$$ \\ $$$$\:\:\:\:\mathrm{x}=\mathrm{10}−\mathrm{log}\left(\mathrm{2}+\mathrm{x}\right)\:\:\:\:\:\mathrm{log100}=\mathrm{2} \\…
Question Number 197388 by cortano12 last updated on 16/Sep/23 Answered by MM42 last updated on 16/Sep/23 $${x}=\mathrm{1}\:\:\&\:\:{y}=\mathrm{1} \\ $$ Answered by Rasheed.Sindhi last updated on…
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Question Number 197099 by Erico last updated on 07/Sep/23 $$\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{ln}\left(\mathrm{cos}{t}\right)}{\mathrm{sin}{t}}\:\mathrm{d}{t}=??? \\ $$ Answered by witcher3 last updated on 07/Sep/23 $$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\left(\mathrm{t}\right)\mathrm{ln}\left(\mathrm{cos}\left(\mathrm{t}\right)\right)}{\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \left(\mathrm{t}\right)}\mathrm{dt},\mathrm{cos}\left(\mathrm{t}\right)=\mathrm{y}…
Question Number 196690 by MATHEMATICSAM last updated on 29/Aug/23 $$\mathrm{If}\:\left({ax}\right)^{\mathrm{log}{a}} \:=\:\left({bx}\right)^{\mathrm{log}{b}} \:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$${x}\:=\:\frac{\mathrm{1}}{{ab}}\:. \\ $$ Answered by BaliramKumar last updated on 29/Aug/23 $$\mathrm{loga}\centerdot\mathrm{log}\left(\mathrm{ax}\right)\:=\:\mathrm{logb}\centerdot\mathrm{log}\left(\mathrm{bx}\right) \\…
Question Number 196683 by MATHEMATICSAM last updated on 29/Aug/23 $$\mathrm{If}\:{y}\:=\:\frac{{e}^{{x}} \:−\:{e}^{−\:{x}} }{{e}^{{x}} \:+\:{e}^{−\:{x}} }\:\mathrm{then}\:\mathrm{show}\:\mathrm{that} \\ $$$${x}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\mathrm{log}_{{e}} \left(\frac{\mathrm{1}\:+\:{y}}{\mathrm{1}\:−\:{y}}\right). \\ $$ Answered by Frix last updated on…
Question Number 196596 by MATHEMATICSAM last updated on 27/Aug/23 $$\mathrm{If}\:{f}\left({x}\right)\:=\:\mathrm{ln}\left(\frac{\mathrm{1}\:+\:{x}}{\mathrm{1}\:−\:{x}}\right)\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$${f}\left(\frac{\mathrm{2}{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\right)\:=\:\mathrm{2}{f}\left({x}\right). \\ $$ Commented by mokys last updated on 27/Aug/23 $${f}\:\left(\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)=\:{ln}\left(\frac{\mathrm{1}+\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }}{\mathrm{1}−\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}}…