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Question Number 54875 by shaddie last updated on 13/Feb/19 $$\mathrm{Given}\:\mathrm{that}\frac{\mathrm{log}\left(\mathrm{3x}+\mathrm{1}\right)^{\mathrm{2x}−\mathrm{1}} }{\mathrm{log}\left(\mathrm{3x}+\mathrm{1}\right)}=\mathrm{5},\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}. \\ $$ Answered by kaivan.ahmadi last updated on 14/Feb/19 $$\mathrm{log}\left(\mathrm{3x}+\mathrm{1}\right)^{\mathrm{2x}−\mathrm{1}} =\mathrm{log}\left(\mathrm{3x}+\mathrm{1}\right)^{\mathrm{5}} \Rightarrow \\ $$$$\left(\mathrm{3x}+\mathrm{1}\right)^{\mathrm{2x}−\mathrm{1}}…
Question Number 120380 by bramlexs22 last updated on 31/Oct/20 $${Given}\:\begin{cases}{\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{10}\right)=\frac{{a}}{{a}−\mathrm{1}}}\\{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{5}\right)=\frac{\mathrm{1}}{{b}}}\end{cases} \\ $$$${find}\:{the}\:{value}\:{of}\:\mathrm{1}+\mathrm{log}\:_{\mathrm{12}} \left(\mathrm{15}\right)\:? \\ $$ Answered by FelipeLz last updated on 31/Oct/20 $$\mathrm{log}_{\mathrm{2}}…
Question Number 120120 by benjo_mathlover last updated on 29/Oct/20 $$\begin{cases}{\mathrm{log}\:_{{a}} \left({x}\right)\:=\:\mathrm{8}}\\{\mathrm{log}\:_{{b}} \left({x}\right)\:=\:\mathrm{3}\:}\\{\mathrm{log}\:_{{c}} \left({x}\right)\:=\:\mathrm{6}}\end{cases}\:\Rightarrow\:\mathrm{log}\:_{{abc}} \:\left({x}\right)=? \\ $$ Answered by bemath last updated on 29/Oct/20 $$\begin{cases}{\mathrm{log}\:_{{a}} \left({x}\right)=\mathrm{8}\Rightarrow\mathrm{log}\:_{{x}}…
Question Number 119802 by bemath last updated on 27/Oct/20 $${Given}\:{a},{b},{c}\:{real}\:{number}\:{and}\:{not}\:{equal}\:{to}\:\mathrm{1}. \\ $$$${If}\:\mathrm{log}\:_{{a}} \left({b}\right)+\mathrm{log}\:_{{b}} \left({c}\right)+\mathrm{log}\:_{{c}} \left({a}\right)=\mathrm{0}\:{then}\: \\ $$$$\left(\mathrm{log}\:_{{a}} \left({b}\right)\right)^{\mathrm{3}} +\left(\mathrm{log}\:_{{b}} \left({c}\right)\right)^{\mathrm{3}} +\left(\mathrm{log}\:_{{c}} \left({a}\right)\right)^{\mathrm{3}} =? \\ $$…
Question Number 53675 by pieroo last updated on 24/Jan/19 $$\mathrm{Given}\:\mathrm{that}\:\mathrm{1}+\mathrm{log}_{\mathrm{3}} \mathrm{x}\:=\mathrm{log}_{\mathrm{27}} \mathrm{y},\:\mathrm{express}\:\mathrm{y} \\ $$$$\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{x}. \\ $$ Answered by kaivan.ahmadi last updated on 24/Jan/19 $$\mathrm{log}_{\mathrm{3}} \mathrm{3}+\mathrm{log}_{\mathrm{3}}…
Question Number 183594 by greougoury555 last updated on 27/Dec/22 $$\:\:\:\:\:\mathrm{log}\:_{\mathrm{0}.\mathrm{5}} \:\sqrt{\mathrm{1}+{x}}\:+\:\mathrm{3log}\:_{\mathrm{0}.\mathrm{25}} \left(\mathrm{1}−{x}\right)=\:\mathrm{log}\:_{\mathrm{1}/\mathrm{16}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}} +\mathrm{2}\: \\ $$ Answered by hmr last updated on 27/Dec/22 $$\bullet\:\mathrm{log}\:_{\mathrm{c}}…
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Question Number 117838 by snipers237 last updated on 13/Oct/20 $$\:\:{Let}\:{be}\:{P}\:\:{the}\:{set}\:{of}\:{prime}\:{numbers}\:{and}\: \\ $$$${A}={P}\cup\left\{\mathrm{0},\mathrm{1}\right\} \\ $$$${Prove}\:{that}\:\:\:\underset{{n}\notin{A}} {\prod}\:\frac{{n}}{\:\sqrt{{n}^{\mathrm{2}} −\mathrm{1}}}\:=\frac{\mathrm{2}}{\pi}\sqrt{\mathrm{3}}\: \\ $$ Answered by mindispower last updated on 14/Oct/20…
Question Number 117826 by snipers237 last updated on 13/Oct/20 $$\:\:{Prove}\:{that}\:{the}\:{Euler}\:{Constant}\:{is}\:{qlso}\:{equal}\:{to} \\ $$$$\underset{{x}\rightarrow−\mathrm{1}} {\mathrm{lim}}\:\:\Gamma\left({x}\right)−\frac{\mathrm{1}}{{x}\left({x}+\mathrm{1}\right)} \\ $$ Answered by mnjuly1970 last updated on 14/Oct/20 $${solution}:: \\ $$$$\:{lim}_{{x}\rightarrow−\mathrm{1}}…