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Question-69665




Question Number 69665 by ozodbek last updated on 26/Sep/19
Commented by ozodbek last updated on 26/Sep/19
solve
$$\mathrm{solve} \\ $$
Answered by mr W last updated on 26/Sep/19
x^y ln x=ln 9  x^y =((ln 9)/(ln x))  yln x=ln (((ln 9)/(ln x)))=ln (ln 9)−ln (ln x)  ⇒y=((ln (ln 9)−ln (ln x))/(ln x))  x^y ln y=ln 7  ((ln 9)/(ln x))ln y=ln 7  ln y=((ln 7 ln x)/(ln 9))  ln ((ln (ln 9)−ln (ln x))/(ln x))=((ln 7 ln x)/(ln 9))  ln [ln (ln 9)−ln (ln x)]−ln (ln x)=(((ln 7)/(ln 9)))ln x  with t=ln x  ln [ln (ln 9)−ln t]−ln t=(((ln 7)/(ln 9)))t  ⇒t=ln x=0.664263  ⇒x=1.943058  ⇒ln y=0.588285  ⇒y=1.800899
$${x}^{{y}} \mathrm{ln}\:{x}=\mathrm{ln}\:\mathrm{9} \\ $$$${x}^{{y}} =\frac{\mathrm{ln}\:\mathrm{9}}{\mathrm{ln}\:{x}} \\ $$$${y}\mathrm{ln}\:{x}=\mathrm{ln}\:\left(\frac{\mathrm{ln}\:\mathrm{9}}{\mathrm{ln}\:{x}}\right)=\mathrm{ln}\:\left(\mathrm{ln}\:\mathrm{9}\right)−\mathrm{ln}\:\left(\mathrm{ln}\:{x}\right) \\ $$$$\Rightarrow{y}=\frac{\mathrm{ln}\:\left(\mathrm{ln}\:\mathrm{9}\right)−\mathrm{ln}\:\left(\mathrm{ln}\:{x}\right)}{\mathrm{ln}\:{x}} \\ $$$${x}^{{y}} \mathrm{ln}\:{y}=\mathrm{ln}\:\mathrm{7} \\ $$$$\frac{\mathrm{ln}\:\mathrm{9}}{\mathrm{ln}\:{x}}\mathrm{ln}\:{y}=\mathrm{ln}\:\mathrm{7} \\ $$$$\mathrm{ln}\:{y}=\frac{\mathrm{ln}\:\mathrm{7}\:\mathrm{ln}\:{x}}{\mathrm{ln}\:\mathrm{9}} \\ $$$$\mathrm{ln}\:\frac{\mathrm{ln}\:\left(\mathrm{ln}\:\mathrm{9}\right)−\mathrm{ln}\:\left(\mathrm{ln}\:{x}\right)}{\mathrm{ln}\:{x}}=\frac{\mathrm{ln}\:\mathrm{7}\:\mathrm{ln}\:{x}}{\mathrm{ln}\:\mathrm{9}} \\ $$$$\mathrm{ln}\:\left[\mathrm{ln}\:\left(\mathrm{ln}\:\mathrm{9}\right)−\mathrm{ln}\:\left(\mathrm{ln}\:{x}\right)\right]−\mathrm{ln}\:\left(\mathrm{ln}\:{x}\right)=\left(\frac{\mathrm{ln}\:\mathrm{7}}{\mathrm{ln}\:\mathrm{9}}\right)\mathrm{ln}\:{x} \\ $$$${with}\:{t}=\mathrm{ln}\:{x} \\ $$$$\mathrm{ln}\:\left[\mathrm{ln}\:\left(\mathrm{ln}\:\mathrm{9}\right)−\mathrm{ln}\:{t}\right]−\mathrm{ln}\:{t}=\left(\frac{\mathrm{ln}\:\mathrm{7}}{\mathrm{ln}\:\mathrm{9}}\right){t} \\ $$$$\Rightarrow{t}=\mathrm{ln}\:{x}=\mathrm{0}.\mathrm{664263} \\ $$$$\Rightarrow{x}=\mathrm{1}.\mathrm{943058} \\ $$$$\Rightarrow\mathrm{ln}\:{y}=\mathrm{0}.\mathrm{588285} \\ $$$$\Rightarrow{y}=\mathrm{1}.\mathrm{800899} \\ $$
Commented by ozodbek last updated on 27/Sep/19
thank you
$$\mathrm{thank}\:\mathrm{you} \\ $$
Commented by otchereabdullai@gmail.com last updated on 29/Sep/19
the great prof W you are truely exceptional
$$\mathrm{the}\:\mathrm{great}\:\mathrm{prof}\:\mathrm{W}\:\mathrm{you}\:\mathrm{are}\:\mathrm{truely}\:\mathrm{exceptional} \\ $$
Commented by mr W last updated on 29/Sep/19
nice to see you back in forum, sir!
$${nice}\:{to}\:{see}\:{you}\:{back}\:{in}\:{forum},\:{sir}! \\ $$

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