Question Number 4399 by alib last updated on 22/Jan/16
$${Solve}\:{for}\:{x} \\ $$$$ \\ $$$${x}^{\mathrm{2}\:{log}\:_{\mathrm{2}} \:{x}} =\mathrm{8} \\ $$
Commented by Rasheed Soomro last updated on 20/Jan/16
$${x}^{\mathrm{2}\:{log}\:_{\mathrm{2}} \:{x}} =\mathrm{8} \\ $$$${x}^{{log}_{\mathrm{2}} {x}^{\mathrm{2}} } =\mathrm{8} \\ $$$${Let}\:\mathrm{8}={x}^{{y}} \\ $$$${log}_{\mathrm{2}} \mathrm{8}={ylog}_{\mathrm{2}} {x} \\ $$$$\mathrm{3}={y}\:{log}_{\mathrm{2}} {x} \\ $$$${y}=\mathrm{3}/{log}_{\mathrm{2}} {x} \\ $$$${So}\:\mathrm{8}={x}^{\mathrm{3}/{log}_{\mathrm{2}} {x}} \\ $$$${x}^{{log}_{\mathrm{2}} {x}^{\mathrm{2}} } ={x}^{\mathrm{3}/{log}_{\mathrm{2}} {x}} \\ $$$${log}_{\mathrm{2}} {x}^{\mathrm{2}} =\mathrm{3}/{log}_{\mathrm{2}} {x} \\ $$$$\left({log}_{\mathrm{2}} {x}^{\mathrm{2}} \right)\left({log}_{\mathrm{2}} {x}\right)=\mathrm{3} \\ $$$$\mathrm{2}\:{log}_{\mathrm{2}} {x}\:{log}_{\mathrm{2}} {x}=\mathrm{3} \\ $$$$\left({log}_{\mathrm{2}} {x}\right)^{\mathrm{2}} =\frac{\mathrm{3}}{\mathrm{2}} \\ $$$${log}_{\mathrm{2}} {x}=\pm\sqrt{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$$${x}=\mathrm{2}^{\pm\sqrt{\mathrm{1}.\mathrm{5}}} \\ $$$${x}\approx\mathrm{2}.\mathrm{3371},\:\mathrm{0}.\mathrm{42787} \\ $$$$ \\ $$
Commented by Yozzii last updated on 20/Jan/16
$${x}^{{log}_{\mathrm{2}} {x}^{\mathrm{2}} } =\mathrm{8} \\ $$$$\left\{{log}_{\mathrm{2}} {x}^{\mathrm{2}} \right\}\left({lnx}\right)={ln}\mathrm{8} \\ $$$$\frac{{lnx}^{\mathrm{2}} }{{ln}\mathrm{2}}=\frac{{ln}\mathrm{8}}{{lnx}}\:\:\left({change}\:{of}\:{base}\right) \\ $$$$\frac{\mathrm{2}{lnx}}{{ln}\mathrm{2}}=\frac{{ln}\mathrm{8}}{{lnx}} \\ $$$$\mathrm{2}\left({lnx}\right)^{\mathrm{2}} ={ln}\mathrm{2}×{ln}\mathrm{8} \\ $$$${lnx}=\pm\sqrt{\frac{{ln}\mathrm{2}×{ln}\mathrm{8}}{\mathrm{2}}} \\ $$$${x}={exp}\left(\pm\sqrt{\mathrm{0}.\mathrm{5}{ln}\mathrm{2}×{ln}\mathrm{8}}\right)\approx\mathrm{2}.\mathrm{3371},\mathrm{0}.\mathrm{4279} \\ $$
Answered by Rasheed Soomro last updated on 22/Jan/16
$${x}^{\mathrm{2}\:{log}\:_{\mathrm{2}} \:{x}} =\mathrm{8} \\ $$$${log}_{\mathrm{2}} \left({x}^{\mathrm{2}{log}_{\mathrm{2}} {x}} \right)={log}_{\mathrm{2}} \mathrm{8} \\ $$$$\mathrm{2}\:{log}_{\mathrm{2}} {x}\:{log}_{\mathrm{2}} {x}=\mathrm{3} \\ $$$$\left({log}_{\mathrm{2}} {x}\right)^{\mathrm{2}} =\mathrm{3}/\mathrm{2} \\ $$$${log}_{\mathrm{2}} {x}=\pm\sqrt{\mathrm{3}/\mathrm{2}} \\ $$$${x}=\mathrm{2}^{\pm\sqrt{\mathrm{1}.\mathrm{5}}} \\ $$$${x}=\mathrm{2}.\mathrm{33714}\:,\:\mathrm{0}.\mathrm{427873} \\ $$