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Solving the Equation (1/9)^x = 729
This article will guide you through solving the equation (1/9)^x = 729. We'll utilize the properties of exponents and logarithms to find the solution for x.
Understanding the Equation
Firstly, let's understand the equation:
- (1/9)^x: This represents 1/9 raised to the power of x.
- 729: This is a constant value.
Our goal is to find the value of x that satisfies this equation.
Expressing Both Sides with the Same Base
To solve this, we need to express both sides of the equation with the same base. Let's express both 1/9 and 729 as powers of 3:
- 1/9 = 3^-2
- 729 = 3^6
Now, our equation becomes:
(3^-2)^x = 3^6
Applying Exponent Rules
Using the rule of exponents that states (a^m)^n = a^(m*n), we can simplify the left side:
3^(-2*x) = 3^6
Equating the Exponents
Now, since both sides have the same base (3), we can equate the exponents:
-2*x = 6
Solving for x
Finally, solving for x:
x = 6 / -2 x = -3
Conclusion
Therefore, the solution to the equation (1/9)^x = 729 is x = -3.