Find x : 625^(x − 5) = 200((√x))^3


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Question Number 41921 by Tawa1 last updated on 15/Aug/18

$Find x : 625^{x - 5} = 200 {(\sqrt{x})}^{3}$
	Answered by tanmay.chaudhury50@gmail.com last updated on 15/Aug/18

	${(5^{4})}^{x - 5} = 5^{2} \times 2^{3} \times x^{\frac{3}{2}}$ $5^{4 x - 20 - 2} = 2^{3} \times x^{\frac{3}{2}}$ $5^{4 x - 22} = 2^{3} \times x^{\frac{3}{2}}$ $\frac{5^{4 x - 22}}{x^{\frac{3}{2}}} = 2^{3}$ $s e e l e f t h a n d s i d e N_{r} = 5^{4 x - 22} t h e l a s t d i g i t o f$ $i t s v a l u e i s e i t h e r 1 o r 5$ $D_{r} = x^{\frac{2}{3}} i f i t s v a l u e i s f (5) t h e n$ $\frac{N_{r}}{D_{r}} = ϕ (5)$ $b u t r i g h t h a n d s i d e 2^{3} . . . f e a s i b i l i t y t o b e c h e c k e d$ $N_{r} = 5^{4 x - 22} d e v i d e d b y a n y n u m b e r c a n n o t y e i l d f (2)$
	Commented by Tawa1 last updated on 15/Aug/18

	$God bless you sir .$
	Commented by Tawa1 last updated on 15/Aug/18

	$But i think lambert function sir MrW use will work$
	Commented by tanmay.chaudhury50@gmail.com last updated on 15/Aug/18

	$i k n o w l a m b a r t f u n c t i o n b u t n e v e r f a c e t h e p r o b l e m$ $i n s c h o o l a n d c o l l e g e l i f e . . . s o i t i s b d t t e r$ ${M r W}_{3} s o l v e i t a n d g o t h r o u g h m y r e a s o n i n g$
	Commented by Tawa1 last updated on 17/Aug/18

	$But you are very good sir . God will help all of you that help us .$
	Answered by MrW3 last updated on 17/Aug/18
	$625^(x−5) =200x^(3/2) 625^((2(x−5))/3) =200^(2/3) x 625^((2x)/3) =625^((2×5)/3) ×200^(2/3) x 5^((8x)/3) =5^((8×5)/3) ×5^(4/3) ×8^(2/3) x 5^((8x)/3) =5^((44)/3) ×8^(2/3) x e^(((8x)/3)×ln x) =5^((44)/3) ×8^(2/3) x 1=5^((44)/3) ×8^(2/3) x×e^(−((8x)/3)×ln x) −(8/3)×ln 5=5^((44)/3) ×8^(2/3) (−((8x)/3)×ln 5)×e^(−((8x)/3)×ln 5) −((8^(1/3) ln 5)/(3×5^((44)/3) ))=(−((8x)/3)×ln 5)×e^(−((8x)/3)×ln 5) ⇒−((8x)/3)×ln 5=W(−((8^(1/3) ln 5)/(3×5^((44)/3) ))) ⇒x=−(3/(8 ln 5))W(−((8^(1/3) ln 5)/(3×5^((44)/3) )))= { ((≈0)),((=6.25)) :}$
	$625^{x - 5} = 200 x^{\frac{3}{2}}$ $625^{\frac{2 (x - 5)}{3}} = 200^{\frac{2}{3}} x$ $625^{\frac{2 x}{3}} = 625^{\frac{2 \times 5}{3}} \times 200^{\frac{2}{3}} x$ $5^{\frac{8 x}{3}} = 5^{\frac{8 \times 5}{3}} \times 5^{\frac{4}{3}} \times 8^{\frac{2}{3}} x$ $5^{\frac{8 x}{3}} = 5^{\frac{44}{3}} \times 8^{\frac{2}{3}} x$ $e^{\frac{8 x}{3} \times \ln x} = 5^{\frac{44}{3}} \times 8^{\frac{2}{3}} x$ $1 = 5^{\frac{44}{3}} \times 8^{\frac{2}{3}} x \times e^{- \frac{8 x}{3} \times \ln x}$ $- \frac{8}{3} \times \ln 5 = 5^{\frac{44}{3}} \times 8^{\frac{2}{3}} (- \frac{8 x}{3} \times \ln 5) \times e^{- \frac{8 x}{3} \times \ln 5}$ $- \frac{8^{\frac{1}{3}} \ln 5}{3 \times 5^{\frac{44}{3}}} = (- \frac{8 x}{3} \times \ln 5) \times e^{- \frac{8 x}{3} \times \ln 5}$ $\Rightarrow - \frac{8 x}{3} \times \ln 5 = W (- \frac{8^{\frac{1}{3}} \ln 5}{3 \times 5^{\frac{44}{3}}})$ $\Rightarrow x = - \frac{3}{8 \ln 5} W (- \frac{8^{\frac{1}{3}} \ln 5}{3 \times 5^{\frac{44}{3}}}) = {\begin{cases} \approx 0 \\ = 6.25 \end{cases}$
	Commented by Tawa1 last updated on 15/Aug/18

	$wow, God bless you sir . x = 0 ? ? ?$
	Commented by MrW3 last updated on 16/Aug/18

	$x \neq 0, b u t a v e r y s m a l l v a l u e .$ $t h e o t h e r s o l u t i o n i s x = 6.25 .$
	Commented by Tawa1 last updated on 16/Aug/18

	$God bless you sir . i appreciate your effort$
	Commented by Tawa1 last updated on 16/Aug/18
	$when i find the lambert of W(− ((ln5)/(3 × 5^((46)/3) ))) i got { ((− 1.02805 × 10^(−11) )),(( − 28.6561)) :} And my x resulted to { ((x = 2.39536 × 10^(−12) )),((x = 6.677)) :} it does not tally with 6.25 because 6.25 is correct . Or i am missing something$
	$when i find the lambert of W (- \frac{\ln 5}{3 \times 5^{\frac{46}{3}}}) i got {\begin{cases} - 1.02805 \times 10^{- 11} \\ - 28.6561 \end{cases}$ $And my x resulted to {\begin{cases} x = 2.39536 \times 10^{- 12} \\ x = 6.677 \end{cases}$ $it does not tally with 6.25 because 6.25 is correct . Or i am missing something$
	Commented by MrW3 last updated on 17/Aug/18

	$p l e a s e s e e m y c o r r e c t i o n . i t s h o u l d b e$ $W (- \frac{8^{\frac{1}{3}} \ln 5}{3 \times 5^{\frac{44}{3}}})$
	Commented by Tawa1 last updated on 17/Aug/18

	$Thanks for your time sir . God bless you .$
	Commented by Tawa1 last updated on 17/Aug/18

	$can lambert W function be used here sir ? ? ? 2^{x} + 3^{x} = 13$
	Commented by Tawa1 last updated on 17/Aug/18

	$if yes . please help .$
	Commented by MrW3 last updated on 17/Aug/18

	$l a m b e r t w f u n c t i o n w a s a l r e a d y u s e d h e r e .$
	Commented by Tawa1 last updated on 17/Aug/18

	$I mean for something like this sir 2^{x} + 3^{x} = 13$
	Commented by Tawa1 last updated on 17/Aug/18

	$I learnt how to find the W from you sir . that is why i can complete the$ $solution . Thanks for your help sir . God bless you .$
	Commented by Tawa1 last updated on 17/Aug/18

	$I mean like this question : find x . 2^{x} + 3^{x} = 13 . can lambert solve it . if yes$ $please show me sir .$
	Commented by MrW3 last updated on 17/Aug/18

	$n o . y o u c a n n o t u s e l a m b e r t t o s o l v e i t .$
	Commented by Tawa1 last updated on 17/Aug/18

	$Oh, thank you sir . Only newton raphson method can work ? ?$
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