Question 233589
6*(3^x)+4*(3^-x)-14=0


3^(-x) is the same as 1/3^x


your equation becomes:


6*(3^x) + 4*(1/3^x) - 14 = 0


multiply both sides of this equation by 3^x to get:


6 * (3^x) * (3^x) + 4 - 14 * (3^x)  = 0


this becomes:


6 * (3^x)^2 - 14 * (3^x) + 4 = 0


Let y = 3^x and your equation becomes:


6*y^2 - 14*y + 4 = 0


divide both sides of this equation by 2 to get:


3*y^2 - 7*y + 2 = 0


You can factor this equation to get:


(3y-1) * (y-2) = 0


This makes:


3y-1 = 0 or y-2 = 0


3y-1 = 0 solves for y to get y = 1/3


y-2 = 0 solves for y to get y = 2


your answers so far are:


y = 1/3 or y = 2


since y = 3^x, this means that:


3^x = 1/3 or 3^x = 2


this is where the logarithms come in.  up to this point it was straight algebra.


work with 3^x = 1/3 first.


take log of both sides to get:


log(3^x) = log(1/3)


this becomes:


x*log(3) = log(1/3)


divide both sides by log(3) to get:


x = log(1/3)/log(3)


this becomes:


x = -1


work with 3^x = 2 next.


take log of both sides to get:


log(3^x) = log(2)


this becomes:


x*log(3) = log(2)


divide both sides by log(3) to get:


x = log(2)/log(3)


this becomes:


x = .630929754


your answer are:


x = -1
or
x = .630929754


you need to confirm these answers are good by plugging into the original equation.


original equation is:


6*(3^x)+4*(3^-x)-14=0


if x = -1, this becomes:


6*(3^(-1))+4*(3^-(-1))-14=0


this becomes:


6/3 + 4*3 - 14 = 0 which becomes 2 + 12 - 14 = 0 which becomes 0 = 0 which is true so x = -1 is good.


if x = .630929754, this becomes:


6*(3^(.630929754))+4*(3^-(.630929754))-14=0


this becomes:


6*2 + 4*(1/2) - 14 = 0 which becomes 12 + 2 - 14 = 0 which becomes 0 = 0 which is true so x = .630929754 is good.