Question 998780

{{{log(5,x^2)-log(5,(x+4))=7}}}


{{{log(5,x^2/(x+4))=7}}}......change the base


{{{log(x^2/(x+4))/log(5)=7}}}


{{{log(x^2/(x+4))=7log(5)}}}


{{{log(x^2/(x+4))=log(5^7)}}}....if log same, than we have


{{{x^2/(x+4)=5^7}}}


{{{x^2/(x+4)=78125}}}


{{{x^2=78125(x+4)}}}


{{{x^2=78125x+312500}}}


{{{x^2-78125x-312500=0}}}.....use quadratic formula


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


{{{x = (-(-78125) +- sqrt( (-78125)^2-4*1*(-312500) ))/(2*1) }}}


{{{x = (78125 +- sqrt( 6103515625+1250000 ))/2 }}}


{{{x = (78125 +- sqrt( 6104765625 ))/2 }}}


{{{x = (78125 +-7625sqrt(105))/2 }}}


solutions:


{{{x = (78125 +7625 sqrt(105))/2 }}}=>{{{x = 78125/2 +7625sqrt(105))/2 }}}


{{{x = (78125 -7625 sqrt(105))/2 }}}=>{{{x = 78125/2 -7625sqrt(105))/2 }}}


the largest value of {{{x}}}  that satisfies is:

{{{x = (78125 +7625 sqrt(105))/2 }}}=>{{{x = 78125/2 +7625sqrt(105))/2 }}}

approximately {{{x=78128.9998}}}