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Solving the Equation (1/5)^x = 25
This article will guide you through solving the equation (1/5)^x = 25. We will explore different methods to find the value of x that satisfies this equation.
Understanding the Equation
The equation (1/5)^x = 25 involves an exponential function where the base is (1/5) and the exponent is x. Our goal is to find the value of x that makes the equation true.
Method 1: Using Logarithms
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Rewrite 25 as a power of (1/5): We know that 25 is the same as (1/5)^-2. This is because (1/5)^-2 = 5^2 = 25.
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Substitute and solve for x: Now our equation becomes: (1/5)^x = (1/5)^-2 Since the bases are the same, we can equate the exponents: x = -2
Therefore, the solution to the equation (1/5)^x = 25 is x = -2.
Method 2: Using Properties of Exponents
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Express both sides with the same base: We can rewrite 25 as 5^2. The equation now becomes: (1/5)^x = 5^2.
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Apply exponent rules: Since (1/5) is the reciprocal of 5, we can write: 5^-x = 5^2
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Equate exponents: Now that both sides have the same base, we can equate the exponents: -x = 2
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Solve for x: Dividing both sides by -1 gives us: x = -2
This method also confirms that the solution to the equation (1/5)^x = 25 is x = -2.
Conclusion
We have successfully solved the equation (1/5)^x = 25 using two different methods: logarithms and properties of exponents. Both methods lead to the same solution: x = -2. This demonstrates the versatility of mathematical tools in solving exponential equations.