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Question Number 80844 by jagoll last updated on 07/Feb/20

$$If\left({\log }_{10}\left(ax\right)\right)\left({\log }_{10}\left(bx\right)\right)=-1$$$$findxintermaandb$$

Commented by jagoll last updated on 07/Feb/20

$$cananyonehelpme$$

Commented by jagoll last updated on 07/Feb/20

$$\frac{b}{a}⩾...$$$$ittheanswer$$

Answered by mr W last updated on 07/Feb/20

$$solutionexistsonlyfora,b>0ora,b<0.$$$$assumea,b>0infollowing.$$$$\log \left(ax\right)\times \log \left(bx\right)=-1$$$$(\log a+\log x)(\log b+\log x)=-1$$$$(A+X)(B+X)=-1$$$$X^2+(A+B)X+(AB+1)=0$$$$X=\frac{-(A+B)\pm \sqrt{{(A-B)}^2-4}}{2}$$$$\log x=\frac{-(\log a+\log b)\pm \sqrt{{(\log a-\log b)}^2-4}}{2}$$$$\log x=\frac{-\log ab\pm \sqrt{{\left(\log \frac{a}{b}\right)}^2-4}}{2}$$$$\Rightarrow x={10}^{\frac{-\log ab\pm \sqrt{{\left(\log \frac{a}{b}\right)}^2-4}}{2}}$$$$$$$${\left(\log \frac{a}{b}\right)}^2⩾4$$$$\log \frac{a}{b}⩽-2or⩾2$$$$\Rightarrow 0<\frac{a}{b}⩽\frac{1}{100}or\frac{a}{b}⩾100$$

Commented by jagoll last updated on 07/Feb/20

$$thankyoumisterW$$

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