Question Number 148300 by mathdanisur last updated on 26/Jul/21
$${lg}^{\mathrm{2}} \left(\mathrm{10}{x}\right)\:+\:{lg}\left(\mathrm{10}{x}\right)\:=\:\mathrm{6}\:-\:{lg}\left({x}\right) \\ $$$${x}\:=\:? \\ $$
Answered by Olaf_Thorendsen last updated on 26/Jul/21
$$\mathrm{lg}^{\mathrm{2}} \left(\mathrm{10}{x}\right)+\mathrm{lg}\left(\mathrm{10}{x}\right)\:=\:\mathrm{6}−\mathrm{lg}\left({x}\right) \\ $$$$\mathrm{lg}^{\mathrm{2}} \left(\mathrm{10}{x}\right)+\mathrm{lg}\left(\mathrm{10}{x}\right)+\mathrm{lg}\left({x}\right)−\mathrm{6}\:=\:\mathrm{0} \\ $$$$\mathrm{lg}^{\mathrm{2}} \left(\mathrm{10}{x}\right)+\mathrm{lg}\left(\mathrm{10}{x}\right)+\mathrm{lg}\left(\mathrm{10}{x}\right)−\mathrm{6}−\mathrm{lg}\left(\mathrm{10}\right)\:=\:\mathrm{0} \\ $$$$\mathrm{lg}^{\mathrm{2}} \left(\mathrm{10}{x}\right)+\mathrm{2lg}\left(\mathrm{10}{x}\right)−\mathrm{7}\:=\:\mathrm{0} \\ $$$$\mathrm{lg}\left(\mathrm{10}{x}\right)\:=\:\frac{−\mathrm{2}\pm\sqrt{\mathrm{4}−\mathrm{4}\left(−\mathrm{7}\right)}}{\mathrm{2}} \\ $$$$\mathrm{lg}\left(\mathrm{10}{x}\right)\:=\:−\mathrm{1}\pm\mathrm{2}\sqrt{\mathrm{2}} \\ $$$$\mathrm{10}{x}\:=\:\mathrm{10}^{−\mathrm{1}\pm\mathrm{2}\sqrt{\mathrm{2}}} \\ $$$${x}\:=\:\mathrm{10}^{−\mathrm{2}\pm\mathrm{2}\sqrt{\mathrm{2}}} \\ $$
Commented by mathdanisur last updated on 27/Jul/21
$${thank}\:{you}\:{Ser} \\ $$
Answered by liberty last updated on 27/Jul/21
$$\Rightarrow\left(\mathrm{1}+\mathrm{log}\:\mathrm{x}\right)^{\mathrm{2}} +\left(\mathrm{1}+\mathrm{log}\:\mathrm{x}\right)=\mathrm{6}−\mathrm{log}\:\mathrm{x} \\ $$$$\Rightarrow\left(\mathrm{1}+\mathrm{t}\right)^{\mathrm{2}} +\mathrm{2t}−\mathrm{5}=\mathrm{0} \\ $$$$\Rightarrow\mathrm{t}^{\mathrm{2}} +\mathrm{4t}−\mathrm{4}=\mathrm{0} \\ $$$$\Rightarrow\left(\mathrm{t}+\mathrm{2}\right)^{\mathrm{2}} \:=\:\mathrm{8} \\ $$$$\Rightarrow\mathrm{t}+\mathrm{2}\:=\:\pm\mathrm{2}\sqrt{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{t}=−\mathrm{2}\pm\mathrm{2}\sqrt{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{log}\:\mathrm{x}=−\mathrm{2}\pm\mathrm{2}\sqrt{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{x}=\mathrm{10}^{−\mathrm{2}\pm\mathrm{2}\sqrt{\mathrm{2}}} \: \\ $$
Commented by mathdanisur last updated on 27/Jul/21
$${thank}\:{you}\:{Ser} \\ $$