Question 86909
{{{(ln( x ))^2 = ln( x^2 )}}}
{{{e^(ln( x ))^2 = x^2}}}
{{{e^(ln( x )*ln( x )) = x^2}}}
{{{(e^ln( x ))^ln( x ) = x^2}}}
{{{x^ln( x ) = x^2}}}
{{{ln( x ) = 2}}}
{{{x = e^2}}}
Simple Way:
{{{(ln( x ))^2 = ln( x^2 )}}}
{{{ln( x )*ln( x ) = ln( x^2 )}}}
{{{ln( x^( ln( x ) )) = ln( x^2 )}}}
{{{x^( ln( x ) ) = x^2}}}
{{{ln( x ) = 2}}}
{{{x = e^2}}}
~Complete Solution~
{{{(ln( x ))^2 = ln( x^2 )}}}
{{{(ln( x ))^2 = 2*ln( x )}}}
{{{ln( x )^2 - 2*ln( x ) = 0}}} Substitute: u = ln( x )
{{{u^2 - 2*u = 0}}}
{{{u(u - 2) = 0}}}
u = 0 and u - 2 = 0
u = 0 and u = 2
ln( x ) = 0 and ln( x ) = 2
x = 1 and x = e^2
{{{graph(400,200,-2,8,-5,5,(ln( x ))^2,ln( x^2 ))}}}