Question and Answers Forum

All Questions      Topic List

Logarithms Questions

Previous in All Question      Next in All Question      

Previous in Logarithms      Next in Logarithms      

Question Number 103047 by bemath last updated on 12/Jul/20

find the product of roots ((2021))^(1/(3 )) x^(log_(2021) (x)) = x^3

$${find}\:{the}\:{product}\:{of}\:{roots}\: \\ $$$$\sqrt[{\mathrm{3}\:}]{\mathrm{2021}}\:{x}^{\mathrm{log}_{\mathrm{2021}} \:\left({x}\right)} \:=\:{x}^{\mathrm{3}} \\ $$

Commented by mr W last updated on 12/Jul/20

Πx=2021^3 ?

$$\Pi{x}=\mathrm{2021}^{\mathrm{3}} \:? \\ $$

Answered by floor(10²Eta[1]) last updated on 12/Jul/20

log_(2021) (x)=y⇒x=2021^y ⇒((2021))^(1/3) .(2021^y )^y =(2021^y )^3 2021^(1/3) .2021^y^2 =2021^(3y) ⇒y^2 +(1/3)=3y⇒y^2 −3y+(1/3)=0 product=x_1 .x_2 =2021^y_1 .2021^y_2 =2021^(y_1 +y_2 ) =2021^3

$$\mathrm{log}_{\mathrm{2021}} \left(\mathrm{x}\right)=\mathrm{y}\Rightarrow\mathrm{x}=\mathrm{2021}^{\mathrm{y}} \\ $$$$\Rightarrow\sqrt[{\mathrm{3}}]{\mathrm{2021}}.\left(\mathrm{2021}^{\mathrm{y}} \right)^{\mathrm{y}} =\left(\mathrm{2021}^{\mathrm{y}} \right)^{\mathrm{3}} \\ $$$$\mathrm{2021}^{\mathrm{1}/\mathrm{3}} .\mathrm{2021}^{\mathrm{y}^{\mathrm{2}} } =\mathrm{2021}^{\mathrm{3y}} \\ $$$$\Rightarrow\mathrm{y}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{3}}=\mathrm{3y}\Rightarrow\mathrm{y}^{\mathrm{2}} −\mathrm{3y}+\frac{\mathrm{1}}{\mathrm{3}}=\mathrm{0} \\ $$$$\mathrm{product}=\mathrm{x}_{\mathrm{1}} .\mathrm{x}_{\mathrm{2}} =\mathrm{2021}^{\mathrm{y}_{\mathrm{1}} } .\mathrm{2021}^{\mathrm{y}_{\mathrm{2}} } \\ $$$$=\mathrm{2021}^{\mathrm{y}_{\mathrm{1}} +\mathrm{y}_{\mathrm{2}} } =\mathrm{2021}^{\mathrm{3}} \\ $$

Answered by bemath last updated on 14/Jul/20

set x=2021^y ⇔ y=log _(2021) (x) ⇒2021^(1/3) (2021^y )^y = 2021^(3y) 2021^(y^2 −3y+(1/3)) = 1 ⇒y^2 −3y+(1/3) = 0 ; Δ=9−(4/3)>0 ⇒(y−a)(y−b) = 0⇒a+b = 3 A=2021^a ;B = 2021^b ⇔A.B = 2021^(a+b) = 2021^3 = 21^3 (mod 100) = 261 (mod 1000)

$${set}\:{x}=\mathrm{2021}^{{y}} \:\Leftrightarrow\:{y}=\mathrm{log}\:_{\mathrm{2021}} \left({x}\right) \\ $$$$\Rightarrow\mathrm{2021}^{\frac{\mathrm{1}}{\mathrm{3}}} \left(\mathrm{2021}^{{y}} \right)^{{y}} \:=\:\mathrm{2021}^{\mathrm{3}{y}} \\ $$$$\mathrm{2021}^{{y}^{\mathrm{2}} −\mathrm{3}{y}+\frac{\mathrm{1}}{\mathrm{3}}} \:=\:\mathrm{1}\: \\ $$$$\Rightarrow{y}^{\mathrm{2}} −\mathrm{3}{y}+\frac{\mathrm{1}}{\mathrm{3}}\:=\:\mathrm{0}\:;\:\Delta=\mathrm{9}−\frac{\mathrm{4}}{\mathrm{3}}>\mathrm{0}\: \\ $$$$\Rightarrow\left({y}−{a}\right)\left({y}−{b}\right)\:=\:\mathrm{0}\Rightarrow{a}+{b}\:=\:\mathrm{3} \\ $$$${A}=\mathrm{2021}^{{a}} \:;{B}\:=\:\mathrm{2021}^{{b}} \\ $$$$\Leftrightarrow{A}.{B}\:=\:\mathrm{2021}^{{a}+{b}} \:=\:\mathrm{2021}^{\mathrm{3}} \\ $$$$=\:\mathrm{21}^{\mathrm{3}} \:\left({mod}\:\mathrm{100}\right)\:=\:\mathrm{261}\:\left({mod}\right. \\ $$$$\left.\mathrm{1000}\right) \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com