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Find the set of solution of the following equation in R $$2 x^3 \left(\log{x}\right)^2 =5 $$ I tried the sub $$x=10^y$$ to be $$\left(10^{3y}\right)y^2=\frac{5}{2}$$ Then i took log to both sides again to be $$3y +2\log{y}=\log{5}-\log{2}$$ But i did not go throw more

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    $\begingroup$ a numerical method will help you! $\endgroup$ – Dr. Sonnhard Graubner Jan 12 '18 at 14:25
  • $\begingroup$ Thank you ,but the solution is required analytical , does it exist ?@Dr.SonnhardGraubner $\endgroup$ – Hussien Mohamed Jan 12 '18 at 14:27
  • $\begingroup$ For $log$ the base is $e$ or $2$? $\endgroup$ – GhD Jan 12 '18 at 14:29
  • $\begingroup$ are you sure that you made no typos? $\endgroup$ – Dr. Sonnhard Graubner Jan 12 '18 at 14:30
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    $\begingroup$ Wolfram Alpha gives $x=2.50499\dots$, and doesn't give any analytic solution $\endgroup$ – John Doe Jan 12 '18 at 15:04
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$x^3(log x)^2=x(xlogx)^2=\frac{5}{2}$

$x(log x^x)^2=\frac{5}{2}$

x can not be equal to one because $log 1=0$ so we may assume $x=\frac{5}{2}=2.5$ . Also $log 2.5^{2.5}≈0.99..≈1$, Therefore $x≈2.5$ can be a solution.

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