To solve this equation, we will first rewrite it as follows:
3^x - 3^(2-x) - 8 = 0
Next, we can use the property of exponents that states 3^(a+b) = 3^a * 3^b, and 3^(a-b) = 3^a / 3^b.
Therefore, we can rewrite the equation as:
3^x - (3^2 3^(-x)) - 8 = 03^x - 9 (1/3^x) - 8 = 03^x - 9/3^x - 8 = 0
Now, let's substitute u = 3^x:
u - 9/u - 8 = 0
Multiply through by u to clear the fraction:
u^2 - 9 - 8u = 0
Rearranging the equation:
u^2 - 8u - 9 = 0
Now, we can factor the quadratic equation:
(u - 9)(u + 1) = 0
Setting each factor to zero:
u - 9 = 0 or u + 1 = 0
u = 9 or u = -1
Since u = 3^x, we have two possible solutions for x:
1) 3^x = 9x = 2
2) 3^x = -1This solution is not valid as we cannot raise a positive number to any power and get a negative result.
Therefore, the only solution to the equation is x = 2.
To solve this equation, we will first rewrite it as follows:
3^x - 3^(2-x) - 8 = 0
Next, we can use the property of exponents that states 3^(a+b) = 3^a * 3^b, and 3^(a-b) = 3^a / 3^b.
Therefore, we can rewrite the equation as:
3^x - (3^2 3^(-x)) - 8 = 0
3^x - 9 (1/3^x) - 8 = 0
3^x - 9/3^x - 8 = 0
Now, let's substitute u = 3^x:
u - 9/u - 8 = 0
Multiply through by u to clear the fraction:
u^2 - 9 - 8u = 0
Rearranging the equation:
u^2 - 8u - 9 = 0
Now, we can factor the quadratic equation:
(u - 9)(u + 1) = 0
Setting each factor to zero:
u - 9 = 0 or u + 1 = 0
u = 9 or u = -1
Since u = 3^x, we have two possible solutions for x:
1) 3^x = 9
x = 2
2) 3^x = -1
This solution is not valid as we cannot raise a positive number to any power and get a negative result.
Therefore, the only solution to the equation is x = 2.