Question Number 74006 by necxxx last updated on 17/Nov/19
$${If}\:\mathrm{3}{x}^{\mathrm{2}} {e}^{\mathrm{log}\:_{{x}} \mathrm{27}} =\mathrm{27000}\:{then}\:{find}\:{x} \\ $$
Commented by necxxx last updated on 17/Nov/19
$${please}\:{how}\:{can}\:{this}\:{be}\:{solved}\:{analytically} \\ $$$${or}\:{even}\:{by}\:{numerical}\:{methods}= \\ $$$${please}\:{help} \\ $$
Answered by MJS last updated on 17/Nov/19
$$\mathrm{3}{x}^{\mathrm{2}} \mathrm{e}^{\frac{\mathrm{ln}\:\mathrm{27}}{\mathrm{ln}\:{x}}} =\mathrm{27000} \\ $$$${x}^{\mathrm{2}} ×\mathrm{27}^{\frac{\mathrm{1}}{\mathrm{ln}\:{x}}} =\mathrm{9000} \\ $$$${x}=\mathrm{e}^{{t}} \\ $$$$\mathrm{27}^{\frac{\mathrm{1}}{{t}}} \mathrm{e}^{\mathrm{2}{t}} =\mathrm{9000} \\ $$$$\mathrm{2}{t}+\frac{\mathrm{3ln}\:\mathrm{3}}{{t}}=\mathrm{ln}\:\mathrm{9000} \\ $$$${t}^{\mathrm{2}} −\frac{\mathrm{ln}\:\mathrm{9000}}{\mathrm{2}}{t}+\frac{\mathrm{3ln}\:\mathrm{3}}{\mathrm{2}}=\mathrm{0} \\ $$$${t}=\frac{\mathrm{ln}\:\mathrm{9000}}{\mathrm{4}}\pm\sqrt{\frac{\left(\mathrm{ln}\:\mathrm{9000}\right)^{\mathrm{2}} }{\mathrm{16}}−\frac{\mathrm{3ln}\:\mathrm{3}}{\mathrm{2}}} \\ $$$${t}_{\mathrm{1}} \approx.\mathrm{396518} \\ $$$${t}_{\mathrm{2}} \approx\mathrm{4}.\mathrm{15597} \\ $$$${x}=\mathrm{e}^{{t}} \\ $$$${x}_{\mathrm{1}} \approx\mathrm{1}.\mathrm{48664} \\ $$$${x}_{\mathrm{2}} \approx\mathrm{63}.\mathrm{8139} \\ $$
Commented by necxxx last updated on 17/Nov/19
$${wow}....{I}\:{really}\:{didnt}\:{think}\:{this}\:{way}. \\ $$$${whats}\:{wrong}\:{with}\:{my}\:{brain}?? \\ $$$${I}'{m}\:{just}\:{so}\:{grateful}\:{sir}\:{MJS} \\ $$
Commented by MJS last updated on 17/Nov/19
$$\mathrm{you}'\mathrm{re}\:\mathrm{welcome} \\ $$$$\mathrm{we}\:\mathrm{all}\:\mathrm{are}\:\mathrm{prejudiced}\:\mathrm{sometimes}\:\mathrm{and}\:\mathrm{cannot} \\ $$$$\mathrm{see}\:\mathrm{the}\:\mathrm{obvious}... \\ $$