Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 45314 by mondodotto@gmail.com last updated on 11/Oct/18

$$\mathbf{solve}\mathbf{for}\mathit{x}$$$${10}^{\mathit{x}}={\mathit{x}}^{50}$$

Answered by MrW3 last updated on 12/Oct/18

$${10}^x=x^{50}$$$$\pm {10}^{\frac{x}{50}}=x$$$$\pm e^{\frac{x}{50}\ln 10}=x$$$${xe}^{-\frac{x}{50}\ln 10}=\pm 1$$$$-\frac{x\ln 10}{50}{e}^{-\frac{x}{50}\ln 10}=\mp \frac{\ln 10}{50}$$$$\Rightarrow -\frac{x\ln 10}{50}=W(\mp \frac{\ln 10}{50})$$$$\Rightarrow x=-\frac{50}{\ln 10}W(\mp \frac{\ln 10}{50})$$$$=\{\begin{array}{l}-\frac{50}{\ln 10}W\left(\frac{\ln 10}{50}\right)=-\frac{50\times 0.044066}{\ln 10}=-0.956881\\ -\frac{50}{\ln 10}W(-\frac{\ln 10}{50})=\{\begin{array}{l}-\frac{50\times (-0.0483321)}{\ln 10}=1.049518\\ -\frac{50\times (-4.605170)}{\ln 10}=100\end{array}\end{array}$$

Commented by mondodotto@gmail.com last updated on 12/Oct/18

$$\mathrm{thanx}\mathrm{atleast}\mathrm{i}\mathrm{get}\mathrm{something}$$

Commented by mondodotto@gmail.com last updated on 12/Oct/18

$$\mathrm{thanx}\mathrm{a}\mathrm{lot}\mathrm{but}\mathrm{i}\mathrm{dont}\mathbf{undstnd}\mathbf{this}\mathbf{method}\mathbf{please}\mathbf{can}\mathbf{you}\mathbf{explain}\mathbf{sir}$$

Commented by MrW3 last updated on 12/Oct/18

$$LambertWfunctionisused.$$$$If{xe}^x=athenx=W\left(a\right).$$

Commented by mondodotto@gmail.com last updated on 12/Oct/18

$$\mathrm{what}\mathrm{is}\mathit{W}$$

Commented by MrW3 last updated on 12/Oct/18

$$ToapplyLambertfunctiontosolve$$$$theequation,allwhatwehavetodo$$$$istoputtheequationintoaformlike$$$${Xe}^X=Y$$$$thenwegetX=\mathbb{W}\left(Y\right).$$

Commented by MrW3 last updated on 12/Oct/18

$$W\left(x\right)isthelambertfunction,defined$$$$asW\left(x\right)e^{W\left(x\right)}=x.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com