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Question Number 175040 by andres_chu last updated on 17/Aug/22

$$$$ hello, please, someone help me to correct the equation? It's typed wrong and I can't find where\n$$Solve:$$ $$5,76[\frac{{\log }_a\left(\sqrt{\log {}_b\left(\sqrt{a}\right)}\right)}{\log \left(\sqrt{\log \left(a\right)}\right)}+{\log }_{\log \left(a\right)}\left(2\right)]\sqrt[5]{\log {}_2\left(x\right)}+\frac{{\log }_2\left(x\right)}{25}={\left[\log {}_2\left(x\right)\right]}^{\frac{3}{5}}$$ $$Answers$$ $$x_1=1,x_2=2^{243},x_3=2^{-243},x_4=2^{1024},x_5=2^{-1024}$$

Answered by a.lgnaoui last updated on 20/Aug/22

$$$$ $$theequationwillbe:$$ $$5,76[\frac{\log (\sqrt{\log (\sqrt{\left(\mathrm{a}\right)}}}{\log (\sqrt{\log \left(\mathrm{a}\right)}}+{\log }_{\log \left(\mathrm{a}\right)}\left(2\right)]{\left(\log {\left(\mathrm{x}\right)}_2\right)}^{1/5}+\frac{{\log }_2\left(\mathrm{x}\right)}{25}={\left({\log }_2\left(\mathrm{x}\right)\right)}^{3/5}$$ $$\frac{\log (\sqrt{\log (\sqrt{\mathrm{a}))}}}{\log (\sqrt{\log \left(\mathrm{a}\right)}}+{\log }_{\log \left(\mathrm{a}\right)}\left(2\right)=[\frac{\frac{1}{2}\log \left(\frac{1}{2}\log \left(\mathrm{a}\right)\right)}{\frac{1}{2}\log \left(\log \left(\mathrm{a}\right)\right)}+\frac{\log \left(2\right)}{\log \left(\log \left(\mathrm{a}\right)\right)}]$$ $$\frac{\log \left(\log \left(\mathrm{a}\right)\right)-\log \left(2\right)}{\log \left(\log \left(\mathrm{a}\right)\right)}+\frac{\log \left(2\right)}{\log \left(\log \left(\mathrm{a}\right)\right)}=1$$ $$5,76{\left({\log }_2\mathrm{x}\right)}^{1/5}+\frac{{\log }_2\left(\mathrm{x}\right)}{25}={\left({\log }_2\left(\mathrm{x}\right)\right)}^{3/5}$$ $$\mathrm{posons}{\left({\log }_2\left(\mathrm{x}\right)\right)}^{1/5}=\mathrm{X}$$ $$\mathrm{X}[5,76+\frac{{\mathrm{X}}^4}{25}-{\mathrm{X}}^2=0]\Rightarrow {\mathrm{X}}^4-{25\mathrm{X}}^2+144=0$$ $$\mathrm{Z}={\mathrm{X}}^2{\mathrm{Z}}^2-25\mathrm{Z}+144=0\mathrm{\Delta }=7^2\mathrm{Z}=\frac{25\pm 7}{2}=(9,16)\mathrm{X}=(3,4)$$ $${\left[{\log }_2\left(\mathrm{x}\right)\right]}^{1/5}=3\Rightarrow \log \left(\mathrm{x}\right)=3^5\log {\left(2\right)}^5\mathrm{x}=7,68\times {10}^{16}$$ $$\mathrm{x}=4{\log }_2\left(\mathrm{x}\right)=4^5[\log \left(\mathrm{x}\right)=4^5\log {\left(2\right)}^5\mathrm{x}=1,428\times {10}^{71}$$

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