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Solving the Equation (1/3)^x = 9
This article will explore how to solve the equation (1/3)^x = 9. We'll use the properties of exponents and logarithms to find the solution for x.
Understanding the Equation
- Fractional Base: The base of the exponent is (1/3), which is a fraction less than 1.
- Exponential Form: The equation is in exponential form, where the unknown x is the exponent.
- Goal: We need to find the value of x that makes the equation true.
Solving the Equation
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Express 9 as a power of (1/3): Since 9 is the square of 3, we can rewrite it as (3)^2. To express 9 as a power of (1/3), remember that (1/3) is the reciprocal of 3. Therefore:
(1/3)^(-2) = 3^2 = 9
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Equate the exponents: Now we have: (1/3)^x = (1/3)^(-2)
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Solve for x: Since the bases are the same, we can equate the exponents: x = -2
Solution
Therefore, the solution to the equation (1/3)^x = 9 is x = -2.
Verification
We can verify our solution by substituting x back into the original equation:
(1/3)^(-2) = (3)^2 = 9
The equation holds true, confirming our solution.
Conclusion
By understanding the properties of exponents and applying them strategically, we can solve exponential equations like (1/3)^x = 9. The solution, x = -2, satisfies the equation and can be verified by substitution.