Solving 5 = 5x² And Verifying Solutions With Logarithms
In the fascinating realm of mathematics, solving equations and verifying solutions through different methods is a cornerstone of understanding. This article delves into the solution of the equation 5 = 5x² and explores the subsequent verification process using logarithms. Our journey will involve algebraic manipulation to find the roots of the equation and logarithmic properties to confirm the validity of these solutions. Through a detailed step-by-step approach, we will illuminate the underlying principles and demonstrate how different mathematical concepts intertwine to produce coherent and logical results. The focus will remain on providing a clear, concise, and comprehensive explanation, making the content accessible to readers of varying mathematical backgrounds. We aim to not only solve the equation but also to enhance the reader's understanding of the processes involved, thereby fostering a deeper appreciation for the beauty and precision of mathematical reasoning. This exploration will showcase the elegance of mathematics, where different branches like algebra and logarithms converge to affirm the correctness of a solution, emphasizing the interconnected nature of mathematical concepts.
Solving the Equation 5 = 5x²
Let's begin by tackling the equation 5 = 5x². To find the values of x that satisfy this equation, we will employ algebraic techniques to isolate x. The initial step involves dividing both sides of the equation by 5. This simplification is crucial as it reduces the coefficients and makes the equation easier to handle. By performing this division, we transform the equation into 1 = x². This form is much more manageable and directly relates x² to a constant value. Next, to solve for x, we take the square root of both sides of the equation. It's essential to remember that taking the square root yields both positive and negative solutions, as squaring either a positive or a negative number results in a positive value. Therefore, taking the square root of 1 gives us ±1, meaning x can be either 1 or -1. These two values, 1 and -1, are the roots of the original equation, and they represent the points where the equation holds true. This part of the solution highlights the fundamental algebraic principles of equation manipulation and the importance of considering both positive and negative roots when dealing with square roots. The process of simplifying and isolating the variable is a common theme in algebra, and this example clearly illustrates how these techniques are applied to find the solution set of a quadratic equation.
Step-by-Step Breakdown
- Initial Equation: 5 = 5x²
- Divide by 5: 1 = x²
- Take the Square Root: √1 = √(x²)
- Solutions: x = ±1
Verifying the Solutions Using Logarithms
Now that we have determined the solutions for the equation 5 = 5x², which are x = 1 and x = -1, we will proceed to verify these solutions using logarithms. This verification process is a crucial step in ensuring the accuracy of our solutions and demonstrates the power of logarithms in confirming algebraic results. The logarithmic verification involves substituting the found values of x back into a logarithmic equation derived from the original problem. Specifically, we are using the expression log₅(5x⁴) - log₅(x²). This expression is chosen because it incorporates the solutions x in a way that, if the solutions are correct, should simplify to a constant value that aligns with the problem's conditions. The key here is to understand how logarithmic properties, such as the quotient rule and the power rule, can be applied to simplify and evaluate logarithmic expressions. When we substitute x = 1 and x = -1 into the expression, we expect the result to be consistent with the given condition, which in this case is that the expression should equal 1. This expectation is based on the understanding that if the solutions are valid, they should satisfy the logarithmic equation just as they satisfy the original algebraic equation. Therefore, this step-by-step substitution and simplification process serves as a robust check on the correctness of our algebraic solutions, demonstrating the interconnectedness of algebraic and logarithmic concepts in mathematics.
Substituting x = 1
To begin, let's substitute x = 1 into the expression log₅(5x⁴) - log₅(x²). Replacing x with 1, we get log₅(5(1)⁴) - log₅((1)²). Simplifying the terms inside the logarithms, we have log₅(5) - log₅(1). Now, we can evaluate these logarithms. The logarithm log₅(5) is equal to 1 because 5 raised to the power of 1 is 5. The logarithm log₅(1) is equal to 0 because any base raised to the power of 0 is 1. Therefore, our expression becomes 1 - 0, which simplifies to 1. This result confirms that x = 1 is indeed a valid solution, as it satisfies the logarithmic equation derived from the original problem. The process highlights how logarithms can be used to verify algebraic solutions by transforming the problem into a different mathematical domain, where different properties and rules can be applied to check for consistency. This step-by-step evaluation demonstrates the practical application of logarithmic properties, such as the understanding of logarithmic values for the base itself and for the number 1, in validating mathematical solutions.
- Substitute x = 1: log₅(5(1)⁴) - log₅((1)²)
- Simplify: log₅(5) - log₅(1)
- Evaluate Logarithms: 1 - 0
- Result: 1
Substituting x = -1
Next, we will substitute x = -1 into the same expression, log₅(5x⁴) - log₅(x²). Replacing x with -1, we obtain log₅(5(-1)⁴) - log₅((-1)²). Simplifying the terms inside the logarithms, we have log₅(5(1)) - log₅(1), which further simplifies to log₅(5) - log₅(1). As we determined in the previous verification step, log₅(5) equals 1, and log₅(1) equals 0. Therefore, the expression becomes 1 - 0, which equals 1. This result also confirms that x = -1 is a valid solution, as it satisfies the logarithmic equation just like x = 1. This verification reinforces the concept that both positive and negative solutions obtained from the original equation are mathematically sound and consistent across different mathematical operations, such as logarithmic evaluations. The successful verification of both solutions using logarithms demonstrates the robustness of our initial algebraic solution and further illustrates the interconnectedness of algebraic and logarithmic principles. By systematically substituting and simplifying, we have shown that both values of x satisfy the given logarithmic condition, thereby solidifying our understanding of the equation and its solutions.
- Substitute x = -1: log₅(5(-1)⁴) - log₅((-1)²)
- Simplify: log₅(5) - log₅(1)
- Evaluate Logarithms: 1 - 0
- Result: 1
Conclusion
In conclusion, we have successfully solved the equation 5 = 5x², finding the solutions x = 1 and x = -1. Furthermore, we have rigorously verified these solutions using logarithms, demonstrating the consistency and validity of our algebraic results. The process involved substituting each solution back into the logarithmic expression log₅(5x⁴) - log₅(x²), and in both cases, the expression simplified to 1, confirming that both x = 1 and x = -1 satisfy the given condition. This comprehensive approach not only solves the equation but also reinforces the understanding of how different mathematical concepts, such as algebra and logarithms, are interconnected. The use of logarithms as a verification tool highlights their power in confirming the accuracy of solutions obtained through algebraic methods. This exploration underscores the importance of a systematic approach to problem-solving in mathematics, where each step is carefully executed and verified. By combining algebraic manipulation with logarithmic verification, we gain a deeper insight into the nature of equations and their solutions, thereby enhancing our mathematical proficiency and appreciation for the subject's coherence and elegance. The process serves as a valuable exercise in mathematical reasoning, demonstrating how multiple tools and techniques can be applied to solve and validate mathematical problems effectively.
