Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 169738 by Mastermind last updated on 07/May/22

$$2^x+x=11$$$$findx?$$$$$$$$Mastermind$$

Commented by mr W last updated on 08/May/22

$$for{2}^x+x=11youcanseethesolution$$$$x=3.foranyothervaluethan11,$$$$forexample{2}^x+x=12,youcannot$$$$seethesolutionanymore,butyou$$$$canstillsolvewithLambertfunction.$$

Answered by Mathspace last updated on 07/May/22

$$f\left(x\right)=2^x+x-11$$$${lim}_{x\to +\mathrm{\infty }}f\left(x\right)=+\mathrm{\infty }$$$${lim}_{x\to -\mathrm{\infty }}f\left(x\right)=-\mathrm{\infty }$$$$f^{{}^{\prime }}\left(x\right)=ln2.2^x+1>0\Rightarrow festcroissante$$$$x-\mathrm{\infty }+\mathrm{\infty }$$$$f^{{}^{\prime }}+$$$$f-\mathrm{\infty }croisscroiss+\mathrm{\infty }$$$$\Rightarrow \mathrm{\exists }one\alpha /f\left(\alpha \right)=0$$$$f\left(3\right)=2^3+3-11=0\Rightarrow \alpha =3$$

Answered by Joshua24 last updated on 07/May/22

$$2^x+x=11$$$$2^x=11-x$$$$f\left(x\right)=2^{x}r(0,\mathrm{\infty })$$$$g\left(x\right)=11-x$$$$2^x\ge 1,11-x\ge 1$$$$0\le x\le 10$$$$2^3+3=11$$$$x=3$$$$$$

Answered by mr W last updated on 07/May/22

$$2^x+x=12$$$$2^x=12-x$$$$2^{12}\times 2^{x-12}=(12-x)$$$$2^{12}\times e^{(x-12)\ln 2}=(12-x)$$$$2^{12}=(12-x)e^{(12-x)\ln 2}$$$$2^{12}\times \ln 2=\left[(12-x)\ln 2\right]e^{(12-x)\ln 2}$$$$(12-x)\ln 2=\mathbb{W}\left(2^{12}\times \ln 2\right)$$$$\Rightarrow x=12-\frac{\mathbb{W}\left(2^{12}\times \ln 2\right)}{\ln 2}$$$$\approx 12-\frac{6.13692882}{\ln 2}=3.146283$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com