4^x + x = 260 find the possible values of x


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Question Number 205238 by necx122 last updated on 13/Mar/24

$4^{x} + x = 260$ $f i n d t h e p o s s i b l e v a l u e s o f x$
Commented by Ghisom last updated on 13/Mar/24

$generally$ $b^{x} = a x + c$ $let x = - \frac{t}{\ln b} - \frac{c}{a}$ $e^{t} t = - \frac{\ln b}{b^{c / a} a}$ $t = W (- \frac{\ln b}{b^{c / a} a})$ $x = - \frac{c}{a} - \frac{1}{\ln b} W (- \frac{\ln b}{b^{c / a} a})$
	Answered by mr W last updated on 13/Mar/24

	$4^{x} = 260 - x$ $4^{x - 260} \times 4^{260} = 260 - x$ $(260 - x) 4^{260 - x} = 4^{260}$ $(260 - x) e^{(260 - x) \ln 4} = 4^{260}$ $(260 - x) \ln 4 e^{(260 - x) \ln 4} = 4^{260} \times \ln 4$ $\Rightarrow (260 - x) \ln 4 = W (4^{260} \times \ln 4)$ $\Rightarrow x = 260 - \frac{W (4^{260} \times \ln 4)}{\ln 4} = 4$
	Commented by necx122 last updated on 13/Mar/24
	This is so clear and understandable. Thank you sir.
	Commented by necx122 last updated on 13/Mar/24

	$M e a n w h i l e, i s t h e L a m b e r t W f u n c t i o n$ $s u p p o s e d t o g i v e u s o t h e r v a l u e s ?$ $H o w d o w e c a l c u l a t e f o r o t h e r s ?$ $W h a t c a l c u l a t o r s d o w e u s e f o r$ $c o m p u t i n g t h e l a m b a r t W f u n c t i o n .$
	Commented by mr W last updated on 13/Mar/24

	$i u s e w o l f r a m a l p h a$
	Commented by mr W last updated on 13/Mar/24

	Commented by mr W last updated on 13/Mar/24

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