To solve this equation, we can rewrite each term with the same base and then solve for x.
27^(x-2/3) = (3^3)^(x-2/3) = 3^(3(x-2/3)) = 3^(3x-2)
9^(x-1) = (3^2)^(x-1) = 3^(2(x-1)) = 3^(2x-2)
Now our equation becomes:
3^(3x-2) - 3^(2x-2) = 2*3^(3x-1)
We can then combine like terms:
3^(3x-2) - 3^(2x-2) = 23^(3x-1)3^(3x-2) - 3^(2x-2) - 23^(3x-1) = 0
Now we have a quadratic equation in terms of 3^(x). Let's make a substitution:
Let y = 3^x
The equation becomes:
y^3 - y^2 - 2y = 0
Factorizing this equation, we get:
y(y^2 - y - 2) = 0y(y - 2)(y + 1) = 0
So, the possible values for y are 0, 2 and -1. However, y cannot be negative.
Therefore, y = 2
Therefore, 3^x = 2
x = log base 3 (2)
x ≈ 0.6309
Therefore, the solution to the equation is x ≈ 0.6309.
To solve this equation, we can rewrite each term with the same base and then solve for x.
27^(x-2/3) = (3^3)^(x-2/3) = 3^(3(x-2/3)) = 3^(3x-2)
9^(x-1) = (3^2)^(x-1) = 3^(2(x-1)) = 3^(2x-2)
Now our equation becomes:
3^(3x-2) - 3^(2x-2) = 2*3^(3x-1)
We can then combine like terms:
3^(3x-2) - 3^(2x-2) = 23^(3x-1)
3^(3x-2) - 3^(2x-2) - 23^(3x-1) = 0
Now we have a quadratic equation in terms of 3^(x). Let's make a substitution:
Let y = 3^x
The equation becomes:
y^3 - y^2 - 2y = 0
Factorizing this equation, we get:
y(y^2 - y - 2) = 0
y(y - 2)(y + 1) = 0
So, the possible values for y are 0, 2 and -1. However, y cannot be negative.
Therefore, y = 2
Therefore, 3^x = 2
x = log base 3 (2)
x ≈ 0.6309
Therefore, the solution to the equation is x ≈ 0.6309.