Question 195566
{{{log(3,(x-5))+log(3,(x+3))=4}}} Start with the given equation.



{{{log(3,((x-5)(x+3)))=4}}} Combine the logs using the identity {{{log(b,(A))+log(b,(B))=log(b,(A*B))}}}



{{{3^4=(x-5)(x+3)}}} Rewrite the equation using the property: {{{log(b,(x))=y}}} ====> {{{b^y=x}}}



{{{81=(x-5)(x+3)}}} Raise 3 to the 4th power to get 81



{{{81=x^2-2x-15}}} FOIL



{{{0=x^2-2x-15-81}}} Subtract 81 from both sides.



{{{0=x^2-2x-96}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-2}}}, and {{{c=-96}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-2) +- sqrt( (-2)^2-4(1)(-96) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-2}}}, and {{{c=-96}}}



{{{x = (2 +- sqrt( (-2)^2-4(1)(-96) ))/(2(1))}}} Negate {{{-2}}} to get {{{2}}}. 



{{{x = (2 +- sqrt( 4-4(1)(-96) ))/(2(1))}}} Square {{{-2}}} to get {{{4}}}. 



{{{x = (2 +- sqrt( 4--384 ))/(2(1))}}} Multiply {{{4(1)(-96)}}} to get {{{-384}}}



{{{x = (2 +- sqrt( 4+384 ))/(2(1))}}} Rewrite {{{sqrt(4--384)}}} as {{{sqrt(4+384)}}}



{{{x = (2 +- sqrt( 388 ))/(2(1))}}} Add {{{4}}} to {{{384}}} to get {{{388}}}



{{{x = (2 +- sqrt( 388 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (2 +- 2*sqrt(97))/(2)}}} Simplify the square root  (note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)  



{{{x = (2)/(2) +- (2*sqrt(97))/(2)}}} Break up the fraction.  



{{{x = 1 +- sqrt(97)}}} Reduce.  



{{{x = 1+sqrt(97)}}} or {{{x = 1-sqrt(97)}}} Break up the expression.  



So the <i>possible</i> answers are {{{x = 1+sqrt(97)}}} or {{{x = 1-sqrt(97)}}} 




However, since you cannot take the log of a negative number, this rules out {{{x = 1-sqrt(97)}}}



So the only solution is {{{x = 1+sqrt(97)}}}