Question Number 68712 by Rio Michael last updated on 15/Sep/19 $${given}\:{that}\:{x}\:{and}\:{y}\:{are}\:{two}\:{numbers}\:{other}\:{one}.\: \\ $$$${given}\:{that}\:\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$$${and}\:\:{a}^{{x}} \:=\:{b}^{{y}} \:=\:\left({ab}\right)^{{xy}} \:\:{show}\:{that}\:\:{x}\:+\:{y}\:=\mathrm{0} \\ $$ Commented by Prithwish sen last…
Question Number 3160 by prakash jain last updated on 06/Dec/15 $$\mathrm{Prove}\:\mathrm{that}\:{e}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}\:\:\mathrm{is}\:\mathrm{irrational}. \\ $$ Answered by Yozzi last updated on 06/Dec/15 $${Fourier}'{s}\:{Proof} \\ $$$${Suppose}\:{for}\:{contradiction}\:{that}…
Question Number 68616 by Rio Michael last updated on 14/Sep/19 $${given}\:{that}\:{a},{b}\:{and}\:{c}\:{are}\:{positive}\:{numbers}\:{other}\:{than}\:\mathrm{1} \\ $$$$,\:{show}\:{that}\:\:{log}_{{b}} {a}\:×\:{log}_{{c}} {b}\:×\:{log}_{{a}} {c}\:=\:\mathrm{1} \\ $$$${hence},\:{evaluate}\:\:\:{log}_{\mathrm{10}} \mathrm{25}\:×\:{log}_{\mathrm{2}} \mathrm{10}\:×\:{log}_{\mathrm{5}} \mathrm{4} \\ $$ Answered by…
Question Number 68618 by Rio Michael last updated on 14/Sep/19 $${solve}\:{for}\:{x}\:{the}\:{following}\:{equations} \\ $$$$\left.{a}\right)\:{log}\:{x}^{\mathrm{3}} \:−\:\mathrm{2}{log}\:{x}^{\mathrm{2}} \:+\:\mathrm{2}{log}\:{x}\:\:+\:\mathrm{2}{log}\:\sqrt{{x}}\:=\:\mathrm{3} \\ $$$$\left.{b}\right)\:{log}_{{x}} \mathrm{24}\:−\mathrm{3}{log}_{{x}} \mathrm{4}\:\:+\:\mathrm{2}{log}_{{x}} \mathrm{3}\:=−\mathrm{3} \\ $$ Answered by Rasheed.Sindhi…
Question Number 134116 by ClarkeMelodyWenkeh last updated on 27/Feb/21 Answered by EDWIN88 last updated on 27/Feb/21 $$\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{x}+\mathrm{3}\right)−\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{x}\right)=\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{8}\right) \\ $$$$\:\Leftrightarrow\:\frac{\mathrm{x}+\mathrm{3}}{\mathrm{x}}\:=\:\mathrm{8}\:,\:\mathrm{3}\:=\:\mathrm{7x}\:,\:\mathrm{x}\:=\:\frac{\mathrm{3}}{\mathrm{7}} \\ $$ Terms…
Question Number 133936 by liberty last updated on 25/Feb/21 $$\mathrm{log}\:_{\mathrm{x}+\mathrm{8}} \left(\mathrm{x}^{\mathrm{2}} −\mathrm{3x}−\mathrm{4}\right)\:<\:\mathrm{2}.\mathrm{log}\:_{\left(\mathrm{4}−\mathrm{x}\right)^{\mathrm{2}} } \left(\mid\mathrm{x}−\mathrm{4}\mid\right)\: \\ $$ Answered by EDWIN88 last updated on 25/Feb/21 $$\:\mathrm{log}\:_{\mathrm{x}+\mathrm{8}} \left(\mathrm{x}^{\mathrm{2}}…
Question Number 133615 by benjo_mathlover last updated on 23/Feb/21 $$\mathrm{If}\:\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{log}\:_{\mathrm{2}} \:\mathrm{x}\right)+\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{log}\:_{\mathrm{4}} \:\mathrm{x}\right)=\mathrm{2} \\ $$$$\mathrm{then}\:\mathrm{log}\:_{\mathrm{5}} \:\sqrt{\mathrm{x}+\sqrt{\mathrm{x}}\:+\mathrm{5}}\:=\:? \\ $$ Answered by TheSupreme last updated on…
Question Number 1880 by alib last updated on 20/Oct/15 $${Solve} \\ $$$$ \\ $$$$\left.\mathrm{1}\right)\:\left({sin}\:\mathrm{2}{x}+\:\sqrt{}\mathrm{3}\:{cos}\:\mathrm{2}{x}\right)^{\mathrm{2}} =\mathrm{2}\:−\mathrm{2}\:{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{3}}−{x}\right) \\ $$$$\left.\mathrm{2}\right)\:{cos}\:{x}\:−\:{cos}\:\mathrm{2}{x}\:=\:{sin}\:\mathrm{3}{x} \\ $$$$ \\ $$$$\left.\mathrm{3}\right)\:{sin}\:\left(\mathrm{15} °+{x}\right)\:+\:{cos}\:\left(\mathrm{45}°+{x}\right)+\:\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0} \\ $$$$\left.\mathrm{4}\right)\:{tan}\:\left(\mathrm{70}°+{x}\right)\:+\:{tan}\:\left(\mathrm{20}°−{x}\right)\:=\:\mathrm{2} \\…
Question Number 67294 by Rio Michael last updated on 25/Aug/19 $${solve}\:{for}\:\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:{the}\:{simultaneous}\:{equation} \\ $$$$\:\mathrm{log}_{\mathrm{3}} {x}\:=\:{y}\:=\:\mathrm{log}\left(\mathrm{2}{x}\:−\:\mathrm{1}\right) \\ $$ Commented by mr W last updated on 25/Aug/19 $$\mathrm{log}_{{a}}…
Question Number 66413 by hmamarques1994@gmail.com last updated on 14/Aug/19 $$\: \\ $$$$\:\:\sqrt{\mathrm{8}+\boldsymbol{\mathrm{log}}_{\mathrm{6}} \left(\boldsymbol{\mathrm{x}}!\right)}+\sqrt{\mathrm{17}−\boldsymbol{\mathrm{log}}_{\boldsymbol{\mathrm{x}}!} \left(\mathrm{6}\right)}\:=\:\mathrm{7} \\ $$$$\: \\ $$ Answered by MJS last updated on 14/Aug/19…