If 3x^2 e^(log _x 27) =27000 then find x


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Question Number 74006 by necxxx last updated on 17/Nov/19

$I f 3 x^{2} e^{\log_{x} 27} = 27000 t h e n f i n d x$
Commented by necxxx last updated on 17/Nov/19

$p l e a s e h o w c a n t h i s b e s o l v e d a n a l y t i c a l l y$ $o r e v e n b y n u m e r i c a l m e t h o d s =$ $p l e a s e h e l p$
	Answered by MJS last updated on 17/Nov/19

	$3 x^{2} e^{\frac{\ln 27}{\ln x}} = 27000$ $x^{2} \times 27^{\frac{1}{\ln x}} = 9000$ $x = e^{t}$ $27^{\frac{1}{t}} e^{2 t} = 9000$ $2 t + \frac{3 \ln 3}{t} = \ln 9000$ $t^{2} - \frac{\ln 9000}{2} t + \frac{3 \ln 3}{2} = 0$ $t = \frac{\ln 9000}{4} \pm \sqrt{\frac{{(\ln 9000)}^{2}}{16} - \frac{3 \ln 3}{2}}$ $t_{1} \approx . 396518$ $t_{2} \approx 4.15597$ $x = e^{t}$ $x_{1} \approx 1.48664$ $x_{2} \approx 63.8139$
	Commented by necxxx last updated on 17/Nov/19

	$w o w . . . . I r e a l l y d i d n t t h i n k t h i s w a y .$ $w h a t s w r o n g w i t h m y b r a i n ? ?$ $I^{'} m j u s t s o g r a t e f u l s i r M J S$
	Commented by MJS last updated on 17/Nov/19

	${you}^{'} re welcome$ $we all are prejudiced sometimes and cannot$ $see the obvious . . .$
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