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If-3x-2-e-log-x-27-27000-then-find-x-




Question Number 74006 by necxxx last updated on 17/Nov/19
If 3x^2 e^(log _x 27) =27000 then find x
$${If}\:\mathrm{3}{x}^{\mathrm{2}} {e}^{\mathrm{log}\:_{{x}} \mathrm{27}} =\mathrm{27000}\:{then}\:{find}\:{x} \\ $$
Commented by necxxx last updated on 17/Nov/19
please how can this be solved analytically  or even by numerical methods=  please help
$${please}\:{how}\:{can}\:{this}\:{be}\:{solved}\:{analytically} \\ $$$${or}\:{even}\:{by}\:{numerical}\:{methods}= \\ $$$${please}\:{help} \\ $$
Answered by MJS last updated on 17/Nov/19
3x^2 e^((ln 27)/(ln x)) =27000  x^2 ×27^(1/(ln x)) =9000  x=e^t   27^(1/t) e^(2t) =9000  2t+((3ln 3)/t)=ln 9000  t^2 −((ln 9000)/2)t+((3ln 3)/2)=0  t=((ln 9000)/4)±(√((((ln 9000)^2 )/(16))−((3ln 3)/2)))  t_1 ≈.396518  t_2 ≈4.15597  x=e^t   x_1 ≈1.48664  x_2 ≈63.8139
$$\mathrm{3}{x}^{\mathrm{2}} \mathrm{e}^{\frac{\mathrm{ln}\:\mathrm{27}}{\mathrm{ln}\:{x}}} =\mathrm{27000} \\ $$$${x}^{\mathrm{2}} ×\mathrm{27}^{\frac{\mathrm{1}}{\mathrm{ln}\:{x}}} =\mathrm{9000} \\ $$$${x}=\mathrm{e}^{{t}} \\ $$$$\mathrm{27}^{\frac{\mathrm{1}}{{t}}} \mathrm{e}^{\mathrm{2}{t}} =\mathrm{9000} \\ $$$$\mathrm{2}{t}+\frac{\mathrm{3ln}\:\mathrm{3}}{{t}}=\mathrm{ln}\:\mathrm{9000} \\ $$$${t}^{\mathrm{2}} −\frac{\mathrm{ln}\:\mathrm{9000}}{\mathrm{2}}{t}+\frac{\mathrm{3ln}\:\mathrm{3}}{\mathrm{2}}=\mathrm{0} \\ $$$${t}=\frac{\mathrm{ln}\:\mathrm{9000}}{\mathrm{4}}\pm\sqrt{\frac{\left(\mathrm{ln}\:\mathrm{9000}\right)^{\mathrm{2}} }{\mathrm{16}}−\frac{\mathrm{3ln}\:\mathrm{3}}{\mathrm{2}}} \\ $$$${t}_{\mathrm{1}} \approx.\mathrm{396518} \\ $$$${t}_{\mathrm{2}} \approx\mathrm{4}.\mathrm{15597} \\ $$$${x}=\mathrm{e}^{{t}} \\ $$$${x}_{\mathrm{1}} \approx\mathrm{1}.\mathrm{48664} \\ $$$${x}_{\mathrm{2}} \approx\mathrm{63}.\mathrm{8139} \\ $$
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
$${wow}….{I}\:{really}\:{didnt}\:{think}\:{this}\:{way}. \\ $$$${whats}\:{wrong}\:{with}\:{my}\:{brain}?? \\ $$$${I}'{m}\:{just}\:{so}\:{grateful}\:{sir}\:{MJS} \\ $$
Commented by MJS last updated on 17/Nov/19
you′re welcome  we all are prejudiced sometimes and cannot  see the obvious...
$$\mathrm{you}'\mathrm{re}\:\mathrm{welcome} \\ $$$$\mathrm{we}\:\mathrm{all}\:\mathrm{are}\:\mathrm{prejudiced}\:\mathrm{sometimes}\:\mathrm{and}\:\mathrm{cannot} \\ $$$$\mathrm{see}\:\mathrm{the}\:\mathrm{obvious}… \\ $$

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