%5Ex*(8%2F27)%5Ex-1%3D2%2F3.jpeg "(9/4)^x*(8/27)^x-1=2/3")
Solving the Exponential Equation: (9/4)^x * (8/27)^(x-1) = 2/3
This article aims to guide you through the process of solving the exponential equation (9/4)^x * (8/27)^(x-1) = 2/3. We will utilize the properties of exponents and logarithms to arrive at the solution.
Simplifying the Equation
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Expressing bases as powers of a common number:
- 9/4 can be written as (3/2)^2
- 8/27 can be written as (2/3)^3
- 2/3 can be written as (2/3)^1
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Substituting these expressions back into the equation: ( (3/2)^2 )^x * ( (2/3)^3 )^(x-1) = (2/3)^1
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Applying power of a power rule: (3/2)^(2x) * (2/3)^(3x-3) = (2/3)^1
Using Properties of Exponents
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Combining terms with the same base: (3/2)^(2x) * (2/3)^(3x-3) = (3/2)^(2x) * (3/2)^(-3x+3) = (3/2)^(2x - 3x + 3) = (3/2)^(-x+3)
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Equating exponents: (3/2)^(-x+3) = (2/3)^1 = (3/2)^(-1)
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Solving for x: -x + 3 = -1 x = 4
Solution
Therefore, the solution to the equation (9/4)^x * (8/27)^(x-1) = 2/3 is x = 4.
Verification
To verify the solution, substitute x = 4 back into the original equation:
(9/4)^4 * (8/27)^(4-1) = (81/16) * (8/27) = (81 * 8) / (16 * 27) = 2/3
The equation holds true, confirming our solution.