# Solve a logarithmic equation 5

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

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

$x^3(log x)^2=x(xlogx)^2=\frac{5}{2}$
$x(log x^x)^2=\frac{5}{2}$
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.