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Question Number 23223 by abwayh last updated on 27/Oct/17

Answered by mrW1 last updated on 27/Oct/17

⇒((ln a)/(ln x))+((ln a)/(ln y))=4×((ln a)/(ln (xy))) ⇒(1/(ln x))+(1/(ln y))=(4/(ln (xy))) ⇒((ln x+ln y)/((ln x)(ln y)))=(4/(ln x+ln y)) ⇒(ln x+ln y)^2 =4(ln x)(ln y) ⇒(ln x−ln y)^2 =0 ⇒ln x=ln y ⇒x=y

$$\Rightarrow\frac{\mathrm{ln}\:\mathrm{a}}{\mathrm{ln}\:\mathrm{x}}+\frac{\mathrm{ln}\:\mathrm{a}}{\mathrm{ln}\:\mathrm{y}}=\mathrm{4}×\frac{\mathrm{ln}\:\mathrm{a}}{\mathrm{ln}\:\left(\mathrm{xy}\right)} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{ln}\:\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{ln}\:\mathrm{y}}=\frac{\mathrm{4}}{\mathrm{ln}\:\left(\mathrm{xy}\right)} \\ $$$$\Rightarrow\frac{\mathrm{ln}\:\mathrm{x}+\mathrm{ln}\:\mathrm{y}}{\left(\mathrm{ln}\:\mathrm{x}\right)\left(\mathrm{ln}\:\mathrm{y}\right)}=\frac{\mathrm{4}}{\mathrm{ln}\:\mathrm{x}+\mathrm{ln}\:\mathrm{y}} \\ $$$$\Rightarrow\left(\mathrm{ln}\:\mathrm{x}+\mathrm{ln}\:\mathrm{y}\right)^{\mathrm{2}} =\mathrm{4}\left(\mathrm{ln}\:\mathrm{x}\right)\left(\mathrm{ln}\:\mathrm{y}\right) \\ $$$$\Rightarrow\left(\mathrm{ln}\:\mathrm{x}−\mathrm{ln}\:\mathrm{y}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$$\Rightarrow\mathrm{ln}\:\mathrm{x}=\mathrm{ln}\:\mathrm{y} \\ $$$$\Rightarrow\mathrm{x}=\mathrm{y} \\ $$

Commented by abwayh last updated on 28/Oct/17

thank you very much,sir

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much},\mathrm{sir} \\ $$

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