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Question Number 4399 by alib last updated on 22/Jan/16

Solve for x x^(2 log _2 x) =8

$${Solve}\:{for}\:{x} \\ $$$$ \\ $$$${x}^{\mathrm{2}\:{log}\:_{\mathrm{2}} \:{x}} =\mathrm{8} \\ $$

Commented by Rasheed Soomro last updated on 20/Jan/16

x^(2 log _2 x) =8 x^(log_2 x^2 ) =8 Let 8=x^y log_2 8=ylog_2 x 3=y log_2 x y=3/log_2 x So 8=x^(3/log_2 x) x^(log_2 x^2 ) =x^(3/log_2 x) log_2 x^2 =3/log_2 x (log_2 x^2 )(log_2 x)=3 2 log_2 x log_2 x=3 (log_2 x)^2 =(3/2) log_2 x=±(√(3/2)) x=2^(±(√(1.5))) x≈2.3371, 0.42787

$${x}^{\mathrm{2}\:{log}\:_{\mathrm{2}} \:{x}} =\mathrm{8} \\ $$$${x}^{{log}_{\mathrm{2}} {x}^{\mathrm{2}} } =\mathrm{8} \\ $$$${Let}\:\mathrm{8}={x}^{{y}} \\ $$$${log}_{\mathrm{2}} \mathrm{8}={ylog}_{\mathrm{2}} {x} \\ $$$$\mathrm{3}={y}\:{log}_{\mathrm{2}} {x} \\ $$$${y}=\mathrm{3}/{log}_{\mathrm{2}} {x} \\ $$$${So}\:\mathrm{8}={x}^{\mathrm{3}/{log}_{\mathrm{2}} {x}} \\ $$$${x}^{{log}_{\mathrm{2}} {x}^{\mathrm{2}} } ={x}^{\mathrm{3}/{log}_{\mathrm{2}} {x}} \\ $$$${log}_{\mathrm{2}} {x}^{\mathrm{2}} =\mathrm{3}/{log}_{\mathrm{2}} {x} \\ $$$$\left({log}_{\mathrm{2}} {x}^{\mathrm{2}} \right)\left({log}_{\mathrm{2}} {x}\right)=\mathrm{3} \\ $$$$\mathrm{2}\:{log}_{\mathrm{2}} {x}\:{log}_{\mathrm{2}} {x}=\mathrm{3} \\ $$$$\left({log}_{\mathrm{2}} {x}\right)^{\mathrm{2}} =\frac{\mathrm{3}}{\mathrm{2}} \\ $$$${log}_{\mathrm{2}} {x}=\pm\sqrt{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$$${x}=\mathrm{2}^{\pm\sqrt{\mathrm{1}.\mathrm{5}}} \\ $$$${x}\approx\mathrm{2}.\mathrm{3371},\:\mathrm{0}.\mathrm{42787} \\ $$$$ \\ $$

Commented by Yozzii last updated on 20/Jan/16

x^(log_2 x^2 ) =8 {log_2 x^2 }(lnx)=ln8 ((lnx^2 )/(ln2))=((ln8)/(lnx)) (change of base) ((2lnx)/(ln2))=((ln8)/(lnx)) 2(lnx)^2 =ln2×ln8 lnx=±(√((ln2×ln8)/2)) x=exp(±(√(0.5ln2×ln8)))≈2.3371,0.4279

$${x}^{{log}_{\mathrm{2}} {x}^{\mathrm{2}} } =\mathrm{8} \\ $$$$\left\{{log}_{\mathrm{2}} {x}^{\mathrm{2}} \right\}\left({lnx}\right)={ln}\mathrm{8} \\ $$$$\frac{{lnx}^{\mathrm{2}} }{{ln}\mathrm{2}}=\frac{{ln}\mathrm{8}}{{lnx}}\:\:\left({change}\:{of}\:{base}\right) \\ $$$$\frac{\mathrm{2}{lnx}}{{ln}\mathrm{2}}=\frac{{ln}\mathrm{8}}{{lnx}} \\ $$$$\mathrm{2}\left({lnx}\right)^{\mathrm{2}} ={ln}\mathrm{2}×{ln}\mathrm{8} \\ $$$${lnx}=\pm\sqrt{\frac{{ln}\mathrm{2}×{ln}\mathrm{8}}{\mathrm{2}}} \\ $$$${x}={exp}\left(\pm\sqrt{\mathrm{0}.\mathrm{5}{ln}\mathrm{2}×{ln}\mathrm{8}}\right)\approx\mathrm{2}.\mathrm{3371},\mathrm{0}.\mathrm{4279} \\ $$

Answered by Rasheed Soomro last updated on 22/Jan/16

x^(2 log _2 x) =8 log_2 (x^(2log_2 x) )=log_2 8 2 log_2 x log_2 x=3 (log_2 x)^2 =3/2 log_2 x=±(√(3/2)) x=2^(±(√(1.5))) x=2.33714 , 0.427873

$${x}^{\mathrm{2}\:{log}\:_{\mathrm{2}} \:{x}} =\mathrm{8} \\ $$$${log}_{\mathrm{2}} \left({x}^{\mathrm{2}{log}_{\mathrm{2}} {x}} \right)={log}_{\mathrm{2}} \mathrm{8} \\ $$$$\mathrm{2}\:{log}_{\mathrm{2}} {x}\:{log}_{\mathrm{2}} {x}=\mathrm{3} \\ $$$$\left({log}_{\mathrm{2}} {x}\right)^{\mathrm{2}} =\mathrm{3}/\mathrm{2} \\ $$$${log}_{\mathrm{2}} {x}=\pm\sqrt{\mathrm{3}/\mathrm{2}} \\ $$$${x}=\mathrm{2}^{\pm\sqrt{\mathrm{1}.\mathrm{5}}} \\ $$$${x}=\mathrm{2}.\mathrm{33714}\:,\:\mathrm{0}.\mathrm{427873} \\ $$

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