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Question Number 100675 by bobhans last updated on 28/Jun/20

$$\mathrm{If}\log {}_{2\mathrm{x}}\left(\frac{1}{18}\right)=\log {}_{18}\left(\frac{1}{3\mathrm{y}}\right)=\log {}_{3\mathrm{y}}\left(\frac{1}{2\mathrm{x}}\right)$$$$\mathrm{find}3\mathrm{x}-2\mathrm{y}$$

Commented by bramlex last updated on 28/Jun/20

$$\iff 2x=3y=18\to \{\begin{array}{l}x=9\\ y=6\end{array}$$$$\therefore 3x-2y=27-12=15$$

Answered by 1549442205 last updated on 28/Jun/20

$$$$$${\log }_{2\mathrm{x}}\left(\frac{1}{18}\right)={\log }_{2\mathrm{x}}\left(1\right)-{\log }_{2\mathrm{x}}18=-{\log }_{2\mathrm{x}}18$$$$={\log }_{3\mathrm{y}}\left(\frac{1}{2\mathrm{x}}\right)={\log }_{3\mathrm{y}}1-{\log }_{3\mathrm{y}}\left(2\mathrm{x}\right)=-{\log }_{3\mathrm{y}}\left(2\mathrm{x}\right)$$$${\log }_{18}\left(\frac{1}{3\mathrm{y}}\right)={\log }_{18}1-{\log }_{18}\left(3\mathrm{y}\right)=-{\log }_{18}\left(3\mathrm{y}\right)$$$$,\mathrm{so}\mathrm{from}\mathrm{the}\mathrm{hypothesis}\mathrm{we}\mathrm{get}:$$$${\log }_{2\mathrm{x}}18={\log }_{3\mathrm{y}}\left(2\mathrm{x}\right)={\log }_{18}\left(3\mathrm{y}\right)=\mathrm{a}.\mathrm{So}$$$$\{\begin{array}{l}{\left(2\mathrm{x}\right)}^{\mathrm{a}}=18\left(1\right)\\ {\left(3\mathrm{y}\right)}^{\mathrm{a}}=2\mathrm{x}\left(2\right)(\ast )\\ {18}^{\mathrm{a}}=3\mathrm{y}\left(3\right)\end{array}$$$$\mathrm{From}\left(2\right)\mathrm{we}\mathrm{get}{\left(2\mathrm{x}\right)}^{\mathrm{a}}={\left[{\left(3\mathrm{y}\right)}^{\mathrm{a}}\right]}^{\mathrm{a}}={\left(3\mathrm{y}\right)}^{{\mathrm{a}}^2}\left(4\right)$$$$\mathrm{From}\left(3\right)\mathrm{we}\mathrm{get}{\left(3\mathrm{y}\right)}^{{\mathrm{a}}^2}={\left({18}^{\mathrm{a}}\right)}^{{\mathrm{a}}^2}={18}^{{\mathrm{a}}^3}\left(5\right)$$$$\mathrm{From}\left(4\right),\left(5\right)\mathrm{we}\mathrm{get}{\left(2\mathrm{x}\right)}^{\mathrm{a}}={18}^{{\mathrm{a}}^3}\left(6\right)$$$$\mathrm{From}\left(1\right)\mathrm{and}\left(6\right)\mathrm{we}\mathrm{obtain}$$$$18={18}^{{\mathrm{a}}^3}\Rightarrow {\mathrm{a}}^3=1\iff \mathrm{a}=1.\mathrm{Replace}\mathrm{into}(\ast )$$$$\mathrm{we}\mathrm{get}\{\begin{array}{l}3\mathrm{y}=2\mathrm{x}\\ 2\mathrm{x}=18\\ 18=3\mathrm{y}\end{array}\iff \{\begin{array}{l}\mathrm{x}=9\\ \mathrm{y}=6\end{array}$$$$\mathrm{Therefore},3\mathbf{x}-2\mathbf{y}=3\times 9-2\times 6=15$$$$$$$$$$$$$$

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