Question 177040
{{{log(12,(x-3)) + log(12,(x+1))=1}}} Start with the given equation.



Take note that {{{x>3}}} (for the first log) and {{{x>-1}}} (for the second log). So the domain is {{{x>3}}} (this interval works for both logs)


Note: remember, you cannot take the log of 0 or a negative number.



{{{log(12,((x-3)(x+1)))=1}}} Combine the logs using the identity {{{log(b,(A))+log(b,(B))=log(b,(A*B))}}}



{{{12^1=(x-3)(x+1)}}} Rewrite the equation using the property: {{{log(b,(x))=y}}} ====> {{{b^y=x}}}



{{{12=(x-3)(x+1)}}} Raise 12 to the first power to get 12



{{{12=x^2-2x-3}}} FOIL



{{{0=x^2-2x-3-12}}} Subtract 12 from both sides.



{{{0=x^2-2x-15}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-2}}}, and {{{c=-15}}}



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)(-15) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-2}}}, and {{{c=-15}}}



{{{x = (2 +- sqrt( (-2)^2-4(1)(-15) ))/(2(1))}}} Negate {{{-2}}} to get {{{2}}}. 



{{{x = (2 +- sqrt( 4-4(1)(-15) ))/(2(1))}}} Square {{{-2}}} to get {{{4}}}. 



{{{x = (2 +- sqrt( 4--60 ))/(2(1))}}} Multiply {{{4(1)(-15)}}} to get {{{-60}}}



{{{x = (2 +- sqrt( 4+60 ))/(2(1))}}} Rewrite {{{sqrt(4--60)}}} as {{{sqrt(4+60)}}}



{{{x = (2 +- sqrt( 64 ))/(2(1))}}} Add {{{4}}} to {{{60}}} to get {{{64}}}



{{{x = (2 +- sqrt( 64 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (2 +- 8)/(2)}}} Take the square root of {{{64}}} to get {{{8}}}. 



{{{x = (2 + 8)/(2)}}} or {{{x = (2 - 8)/(2)}}} Break up the expression. 



{{{x = (10)/(2)}}} or {{{x =  (-6)/(2)}}} Combine like terms. 



{{{x = 5}}} or {{{x = -3}}} Simplify. 



So the possible answers are {{{x = 5}}} or {{{x = -3}}}



However, since the value {{{x = -3}}} is NOT in the domain {{{x>3}}}, this means that {{{x = -3}}} is NOT a valid solution.



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Answer:



So the only solution is {{{x = 5}}}