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

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

$$\mathrm{4}^{{x}} \:+\:{x}\:=\:\mathrm{260} \\ $$$${find}\:{the}\:{possible}\:{values}\:{of}\:{x} \\ $$$$ \\ $$

Commented by Ghisom last updated on 13/Mar/24

generally b^x =ax+c let x=−(t/(ln b))−(c/a) e^t t=−((ln b)/(b^(c/a) a)) t=W (−((ln b)/(b^(c/a) a))) x=−(c/a)−(1/(ln b))W (−((ln b)/(b^(c/a) a)))

$$\mathrm{generally} \\ $$$${b}^{{x}} ={ax}+{c} \\ $$$$\mathrm{let}\:{x}=−\frac{{t}}{\mathrm{ln}\:{b}}−\frac{{c}}{{a}} \\ $$$$\mathrm{e}^{{t}} {t}=−\frac{\mathrm{ln}\:{b}}{{b}^{{c}/{a}} {a}} \\ $$$${t}={W}\:\left(−\frac{\mathrm{ln}\:{b}}{{b}^{{c}/{a}} {a}}\right) \\ $$$${x}=−\frac{{c}}{{a}}−\frac{\mathrm{1}}{\mathrm{ln}\:{b}}{W}\:\left(−\frac{\mathrm{ln}\:{b}}{{b}^{{c}/{a}} {a}}\right) \\ $$

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

4^x =260−x 4^(x−260) ×4^(260) =260−x (260−x)4^(260−x) =4^(260) (260−x)e^((260−x)ln 4) =4^(260) (260−x)ln 4e^((260−x)ln 4) =4^(260) ×ln 4 ⇒(260−x)ln 4=W(4^(260) ×ln 4) ⇒x=260−((W(4^(260) ×ln 4))/(ln 4))=4

$$\mathrm{4}^{{x}} =\mathrm{260}−{x} \\ $$$$\mathrm{4}^{{x}−\mathrm{260}} ×\mathrm{4}^{\mathrm{260}} =\mathrm{260}−{x} \\ $$$$\left(\mathrm{260}−{x}\right)\mathrm{4}^{\mathrm{260}−{x}} =\mathrm{4}^{\mathrm{260}} \\ $$$$\left(\mathrm{260}−{x}\right){e}^{\left(\mathrm{260}−{x}\right)\mathrm{ln}\:\mathrm{4}} =\mathrm{4}^{\mathrm{260}} \\ $$$$\left(\mathrm{260}−{x}\right)\mathrm{ln}\:\mathrm{4}{e}^{\left(\mathrm{260}−{x}\right)\mathrm{ln}\:\mathrm{4}} =\mathrm{4}^{\mathrm{260}} ×\mathrm{ln}\:\mathrm{4} \\ $$$$\Rightarrow\left(\mathrm{260}−{x}\right)\mathrm{ln}\:\mathrm{4}={W}\left(\mathrm{4}^{\mathrm{260}} ×\mathrm{ln}\:\mathrm{4}\right) \\ $$$$\Rightarrow{x}=\mathrm{260}−\frac{{W}\left(\mathrm{4}^{\mathrm{260}} ×\mathrm{ln}\:\mathrm{4}\right)}{\mathrm{ln}\:\mathrm{4}}=\mathrm{4} \\ $$

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

Meanwhile, is the Lambert W function supposed to give us other values? How do we calculate for others? What calculators do we use for computing the lambart W function.

$${Meanwhile},\:{is}\:{the}\:{Lambert}\:{W}\:{function} \\ $$$${supposed}\:{to}\:{give}\:{us}\:{other}\:{values}? \\ $$$${How}\:{do}\:{we}\:{calculate}\:{for}\:{others}? \\ $$$${What}\:{calculators}\:{do}\:{we}\:{use}\:{for} \\ $$$${computing}\:{the}\:{lambart}\:{W}\:{function}. \\ $$

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

i use wolframalpha

$${i}\:{use}\:{wolframalpha} \\ $$

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

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

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