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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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