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Question Number 32912 by mondodotto@gmail.com last updated on 06/Apr/18

$$\mathbf{find}\mathbf{x}$$$$\mathbf{log}\sqrt{\mathit{x}}=\sqrt{\mathbf{log}\mathit{x}}$$

Answered by mrW2 last updated on 06/Apr/18

$$x⩾0$$$$\log x⩾0\Rightarrow x⩾1$$$$\frac{1}{2}\log x=\sqrt{\log x}$$$$\Rightarrow \frac{1}{4}{\left(\log x\right)}^2=\log x$$$$\Rightarrow (\frac{1}{4}\log x-1)\log x=0$$$$\Rightarrow \log x=0\Rightarrow x=1$$$$\Rightarrow \frac{1}{4}\log x-1=0\Rightarrow \log x=4\Rightarrow x={10}^4$$

Commented by mondodotto@gmail.com last updated on 06/Apr/18

$$\mathrm{thanks}$$

Commented by MJS last updated on 06/Apr/18

$$\mathrm{sometimes}\log \mathrm{means}\ln ,\mathrm{everything}$$$$\mathrm{stays}\mathrm{the}\mathrm{same}\mathrm{in}\mathrm{this}\mathrm{case},\mathrm{but}$$$$\mathrm{the}{2}^{\mathrm{nd}}\mathrm{solution}\mathrm{would}\mathrm{be}x=e^4$$

Commented by mondodotto@gmail.com last updated on 06/Apr/18

$$\mathbf{still}\mathbf{i}{\mathbf{don}}^{\prime }\mathbf{t}\mathbf{get}\mathbf{you},\mathbf{more}\mathbf{explaination}\mathbf{please}$$

Commented by MJS last updated on 06/Apr/18

$$\mathrm{everything}\mathrm{stays}\mathrm{the}\mathrm{same}\mathrm{with}$$$$\mathrm{logarithm}\mathrm{to}\mathrm{base}b,\mathrm{except}\mathrm{the}$$$$\mathrm{last}\mathrm{line}:$$$$\frac{1}{4}{\log }_{b}x-1=0\Rightarrow {\log }_{b}x=4\Rightarrow$$$$\Rightarrow x=b^4$$$$\mathrm{if}\log ={\log }_{10}\Rightarrow x={10}^4$$$$\mathrm{if}\log =\ln ={\log }_e\Rightarrow x=e^4$$

Commented by mondodotto@gmail.com last updated on 07/Apr/18

$$\mathrm{thanx}$$

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