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Solve-for-x-x-2-log-2-x-8-




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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