Question 214854
{{{9^(x-1)-3^(x-1)-2=0}}} Start with the given equation.



{{{(9^x)/9^1-(3^x)/3^1-2=0}}} Rewrite the left side using the identity {{{x^(y-z)=(x^(y))/(x^(z))}}}



{{{(9^x)/9-(3^x)/3-2=0}}} Simplify



{{{cross(9)((9^x)/cross(9))-cross(9)^3((3^x)/cross(3))-9(2)=9(0)}}} Multiply EVERY term by the LCD 9 to clear out the fractions.



{{{9^x-3*3^x-18=0}}} Simplify



{{{(3^2)^x-3*3^x-18=0}}} Rewrite 9 as {{{3^2}}}



{{{3^(2x)-3*3^x-18=0}}} Multiply the exponents.



{{{(3^x)^2-3*3^x-18=0}}} Factor out the exponent 2 using the identity {{{x^(y*z)=(x^(z))^y}}}



Now let's make a substitution to make things easier on us. Let {{{z=3^x}}} (since there are 2 copies of {{{3^x}}})



{{{z^2-3z-18=0}}} Replace each {{{3^x}}} with 'z'



Notice that the quadratic {{{z^2-3z-18}}} is in the form of {{{Az^2+Bz+C}}} where {{{A=1}}}, {{{B=-3}}}, and {{{C=-18}}}



Let's use the quadratic formula to solve for "z":



{{{z = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{z = (-(-3) +- sqrt( (-3)^2-4(1)(-18) ))/(2(1))}}} Plug in  {{{A=1}}}, {{{B=-3}}}, and {{{C=-18}}}



{{{z = (3 +- sqrt( (-3)^2-4(1)(-18) ))/(2(1))}}} Negate {{{-3}}} to get {{{3}}}. 



{{{z = (3 +- sqrt( 9-4(1)(-18) ))/(2(1))}}} Square {{{-3}}} to get {{{9}}}. 



{{{z = (3 +- sqrt( 9--72 ))/(2(1))}}} Multiply {{{4(1)(-18)}}} to get {{{-72}}}



{{{z = (3 +- sqrt( 9+72 ))/(2(1))}}} Rewrite {{{sqrt(9--72)}}} as {{{sqrt(9+72)}}}



{{{z = (3 +- sqrt( 81 ))/(2(1))}}} Add {{{9}}} to {{{72}}} to get {{{81}}}



{{{z = (3 +- sqrt( 81 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{z = (3 +- 9)/(2)}}} Take the square root of {{{81}}} to get {{{9}}}. 



{{{z = (3 + 9)/(2)}}} or {{{z = (3 - 9)/(2)}}} Break up the expression. 



{{{z = (12)/(2)}}} or {{{z =  (-6)/(2)}}} Combine like terms. 



{{{z = 6}}} or {{{z = -3}}} Simplify. 



So the solutions are {{{z = 6}}} or {{{z = -3}}} 



Recall that we let {{{z=3^x}}}. Since {{{3^x>0}}} for all 'x', the solution {{{z=-3}}} isn't possible (since {{{3^x<>-3}}})

  

So we're only going to focus on {{{z = 6}}}



{{{z=3^x}}} Start with the given equation.



{{{6=3^x}}} Plug in {{{z = 6}}}



{{{log(3,(6))=x}}} Convert to logarithmic form.



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Answer:



So the only solution is {{{x=log(3,(6))}}}