Solving the Equation: 5^(2x+1) = 10^(2x+1)
This article provides a comprehensive guide on how to solve the equation 5^(2x+1) = 10^(2x+1). We'll explore various methods, focusing on clear explanations and step-by-step solutions. Understanding logarithmic properties is crucial for efficiently solving exponential equations like this.
Understanding the Equation
The equation 5^(2x+1) = 10^(2x+1) presents an exponential equation where the bases (5 and 10) and exponents (2x+1) are different. Our goal is to find the value of 'x' that satisfies this equality.
Method 1: Using Logarithms
This is the most straightforward method. We'll take the logarithm of both sides of the equation. Using the natural logarithm (ln) simplifies the process:
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Take the natural logarithm of both sides: ln(5^(2x+1)) = ln(10^(2x+1))
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Apply the power rule of logarithms: (2x+1)ln(5) = (2x+1)ln(10)
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Notice that (2x+1) is a common factor. We can simplify: (2x+1)[ln(5) - ln(10)] = 0
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Solve for x:
- Case 1: 2x + 1 = 0 => 2x = -1 => x = -1/2
- Case 2: ln(5) - ln(10) = 0. This case simplifies to ln(5/10) = ln(1/2) = 0, which is not true. Therefore, this case yields no solution.
Therefore, the only solution to the equation is x = -1/2.
Verification
Let's verify our solution by substituting x = -1/2 back into the original equation:
5^(2(-1/2)+1) = 5^0 = 1 10^(2(-1/2)+1) = 10^0 = 1
Since both sides equal 1, our solution x = -1/2 is correct.
Method 2: Direct Comparison (Less Common Approach)
While less common, a direct comparison can be insightful:
Since the exponents are identical, we can consider the bases:
5^(2x+1) = 10^(2x+1) implies 5 = 10 (only if the exponent is zero). However this isn't true in this scenario. This approach alone won't help solve this particular equation unless you recognize that the only time two different numbers raised to the same power are equal is when that power is zero. This leads us directly to solving 2x + 1 = 0, which gives us x = -1/2.
Conclusion
Solving exponential equations like 5^(2x+1) = 10^(2x+1) effectively utilizes the properties of logarithms. The logarithmic approach provides a systematic and reliable method for determining the value of x. Remember to always verify your solution by substituting it back into the original equation. This comprehensive guide provides you with the knowledge and steps to effectively solve similar equations in the future. Understanding logarithmic properties is key to mastering these types of problems.
