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2-x-5-3-y-9-and-25-z-8-find-x-y-z-




Question Number 146532 by mathdanisur last updated on 13/Jul/21
2^x  = 5  ,  3^y  = 9  and  25^z  = 8  find   (x∙y∙z)=?
$$\mathrm{2}^{\boldsymbol{{x}}} \:=\:\mathrm{5}\:\:,\:\:\mathrm{3}^{\boldsymbol{{y}}} \:=\:\mathrm{9}\:\:{and}\:\:\mathrm{25}^{\boldsymbol{{z}}} \:=\:\mathrm{8} \\ $$$${find}\:\:\:\left({x}\centerdot{y}\centerdot{z}\right)=? \\ $$
Answered by puissant last updated on 13/Jul/21
e^(xln2) =5 ⇒ x=((ln5)/(ln2))  e^(yln3) =9 ⇒ y=((2ln3)/(ln3))=2  e^(zln25) =8 ⇒ z=((3ln2)/(2ln5))  (x∙y∙z)=((ln5)/(ln2))×((3ln2)/(2ln5))×2 = 3..
$$\mathrm{e}^{\mathrm{xln2}} =\mathrm{5}\:\Rightarrow\:\mathrm{x}=\frac{\mathrm{ln5}}{\mathrm{ln2}} \\ $$$$\mathrm{e}^{\mathrm{yln3}} =\mathrm{9}\:\Rightarrow\:\mathrm{y}=\frac{\mathrm{2ln3}}{\mathrm{ln3}}=\mathrm{2} \\ $$$$\mathrm{e}^{\mathrm{zln25}} =\mathrm{8}\:\Rightarrow\:\mathrm{z}=\frac{\mathrm{3ln2}}{\mathrm{2ln5}} \\ $$$$\left(\mathrm{x}\centerdot\mathrm{y}\centerdot\mathrm{z}\right)=\frac{\mathrm{ln5}}{\mathrm{ln2}}×\frac{\mathrm{3ln2}}{\mathrm{2ln5}}×\mathrm{2}\:=\:\mathrm{3}.. \\ $$
Commented by mathdanisur last updated on 13/Jul/21
cool thanks Ser
$${cool}\:{thanks}\:{Ser} \\ $$
Answered by Rasheed.Sindhi last updated on 14/Jul/21
2^x  = 5  ,  3^y  = 9  and  25^z  = 8   x∙y∙z=?                                                                  _(−)   25^z  = 8⇒5^(2z) =8⇒(2^x )^(2z) =2^3   ⇒2xz=3⇒xz=(3/2)....(i)  3^y =9⇒3^y =3^2 ⇒y=2.............(ii)  (i)×(ii): xyz=((3/2))(2)=3
$$\mathrm{2}^{\boldsymbol{{x}}} \:=\:\mathrm{5}\:\:,\:\:\mathrm{3}^{\boldsymbol{{y}}} \:=\:\mathrm{9}\:\:{and}\:\:\mathrm{25}^{\boldsymbol{{z}}} \:=\:\mathrm{8} \\ $$$$\underset{−} {\:{x}\centerdot{y}\centerdot{z}=?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} \\ $$$$\mathrm{25}^{\boldsymbol{{z}}} \:=\:\mathrm{8}\Rightarrow\mathrm{5}^{\mathrm{2}{z}} =\mathrm{8}\Rightarrow\left(\mathrm{2}^{{x}} \right)^{\mathrm{2}{z}} =\mathrm{2}^{\mathrm{3}} \\ $$$$\Rightarrow\mathrm{2}{xz}=\mathrm{3}\Rightarrow{xz}=\frac{\mathrm{3}}{\mathrm{2}}….\left({i}\right) \\ $$$$\mathrm{3}^{{y}} =\mathrm{9}\Rightarrow\mathrm{3}^{{y}} =\mathrm{3}^{\mathrm{2}} \Rightarrow{y}=\mathrm{2}………….\left({ii}\right) \\ $$$$\left({i}\right)×\left({ii}\right):\:{xyz}=\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\left(\mathrm{2}\right)=\mathrm{3} \\ $$
Commented by mathdanisur last updated on 14/Jul/21
thanks Ser cool
$${thanks}\:{Ser}\:{cool} \\ $$

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