Question 664304
{{{Ln(x)+Ln(x+2)= -3}}}

{{{Ln(x(x+2))= -3}}}...rewrite as an exponential:

Remember that if log base {{{e}}} of {{{y = x}}}, then {{{e^x = y}}}.

{{{Ln(x^2+2x)= -3}}}

{{{x^2 + 2x=e^-3}}}

{{{x^2 + 2x=1/e^3}}}

 {{{x^2 + 2x + 1/e^3=0}}}

 now use the quadratic formula and eject any answer that makes {{{x}}} come out negative since you can't do log of a negative:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}


{{{x = (-2 +- sqrt( 2^2-4*1*(1/e^3) ))/(2*1) }}}

{{{x = (-2 +- sqrt( 4-4/e^3 ))/2 }}}


{{{x = (-2 +- sqrt( (4e^3-4)/e^3 ))/2 }}}


{{{x = (-2 +- sqrt( 4(e^3-1)/e^3 ))/2 }}}


{{{x = (-2 +- 2)sqrt( (e^3-1)/e^3 ))/2 }}}


{{{x = (-cross(2) +- (cross(2))sqrt( (e^3-1)/e^3 ))/cross(2) }}}


{{{x = -1 +- sqrt( (e^3-1)/e^3)) }}}..take a positive root


{{{x = -1+ sqrt( (e^3-1)/e^3)) }}}...or


{{{x = sqrt( (e^3-1)/e^3))-1 }}}