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Question Number 172111 by Mikenice last updated on 23/Jun/22

$$solve$$$$2^x=4x$$

Commented by mr W last updated on 23/Jun/22

$$youhaveasked{2}^x=10x.itissolved$$$$generallyfor{a}^x=bx.soplease{don}^{\prime }t$$$$postsimilarquestionslike$$$$2^x=4x,$$$$2^x=5xetc...$$

Answered by Mikenice last updated on 23/Jun/22

$$let{2}^x=p$$$$\Rightarrow x=\sqrt{p}$$$$p=4\sqrt{p}$$$$takesquaretobothside$$$$p^2=16p$$$$p=16,then$$$$2^x=16$$$$2^x=2^4$$$$x=4.$$$$$$$$$$

Commented by mr W last updated on 23/Jun/22

$$x=0.3099issolutiontoo.$$

Commented by mr W last updated on 23/Jun/22

$$pleasecheckfollowingstepsir:$$$$let{2}^x=p$$$$\Rightarrow x=\sqrt{p}shoulditnotbex={\log }_{2}p?$$

Commented by Mikenice last updated on 23/Jun/22

$$no,thatiswhatitwillbebutyoucantryitbyusingin$$$$yourownway$$

Commented by mr W last updated on 24/Jun/22

$$2^x=p\Rightarrow x=\sqrt{p}iswrong.$$$$notisaythat,butthemathematics$$$$saysthat.$$$$ITISNOTPERSONALsir.$$

Answered by Mikenice last updated on 24/Jun/22

$$let{2}^x=y---1$$$$x={log}_{2}y---2$$$$equateeqn2byeqn1$$$$y=4{log}_{2}y,thendividebothsideby4$$$$\frac{y}{4}={log}_{2}y\Rightarrow y=2^{\frac{y}{4}}---3$$$$subeqn3intoeqn1$$$$2^x=2^{\frac{y}{4}}$$$$x=\frac{y}{4}---4$$$$lety=16,then$$$$x=\frac{16}{4}\Rightarrow x=4.$$

Answered by mr W last updated on 25/Jun/22

$$2^x=4x$$$$e^{x\ln 2}=4x$$$$(-x\ln 2)e^{-x\ln 2}=-\frac{\ln 2}{4}$$$$-x\ln 2=W(-\frac{\ln 2}{4})$$$$\Rightarrow x=-\frac{W(-\frac{\ln 2}{4})}{\ln 2}$$$$=\{\begin{array}{l}-\frac{-2.77258872}{\ln 2}=4\\ -\frac{-0.21481112}{\ln 2}=0.309907\end{array}$$$$$$$$generallyfor$$$${\mathit{a}}^{\mathit{x}}=\mathit{b}\mathit{x}$$$$\Rightarrow \mathit{x}=-\frac{\mathit{W}(-\frac{\mathbf{ln}\mathit{a}}{\mathit{b}})}{\mathbf{ln}\mathit{a}}$$

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