Find The Equation With X=4 Solution Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Equations form the backbone of various mathematical disciplines, from algebra to calculus, and are essential for understanding and modeling real-world phenomena. When we talk about finding a solution to an equation, we're essentially looking for the value (or values) of the variable that makes the equation true. In this article, we will dive deep into logarithmic equations and explore how to determine which equation has x=4 as its solution. This involves understanding the properties of logarithms, converting logarithmic forms to exponential forms, and verifying the solutions. So, let's embark on this mathematical journey together and unravel the mystery behind these equations!

Understanding Logarithmic Equations

Logarithmic equations might seem daunting at first, but they are quite manageable once you grasp the basics. A logarithm is essentially the inverse operation to exponentiation. To put it simply, if we have an exponential expression like b**y = x, we can express this relationship in logarithmic form as logb(x) = y. Here, b is the base of the logarithm, x is the argument, and y is the exponent. Understanding this fundamental relationship is crucial for solving logarithmic equations effectively. Logarithmic equations come in various forms, and the key to solving them lies in converting them into their equivalent exponential forms. This conversion allows us to eliminate the logarithm and solve for the unknown variable using standard algebraic techniques. It's also important to remember that the base of a logarithm must be a positive number not equal to 1, and the argument must be positive. These conditions are crucial for the logarithm to be defined. Furthermore, familiarity with the properties of logarithms, such as the product rule, quotient rule, and power rule, can greatly simplify the process of solving complex logarithmic equations. These rules allow us to manipulate logarithmic expressions, combine or separate logarithms, and ultimately isolate the variable we are trying to solve for. Therefore, a solid grasp of logarithmic principles is essential for successfully tackling any logarithmic equation.

Analyzing the Equations

Now, let's delve into the specific equations presented and systematically determine which one has x=4 as a solution. We will analyze each equation individually, applying our understanding of logarithms and algebraic techniques. The first equation we encounter is log₄(3x+4) = 2. To solve this equation, we will convert it to its exponential form. Remember, the logarithmic equation logb(x) = y is equivalent to the exponential equation b**y = x. Applying this to our equation, we get 4² = 3x + 4. Simplifying this, we have 16 = 3x + 4. We can now solve for x by subtracting 4 from both sides, giving us 12 = 3x. Dividing both sides by 3, we find x = 4. So, it appears that the first equation has x=4 as a potential solution. We will confirm this later by substituting x=4 back into the original equation. The second equation is log₃(2x-5) = 2. Similar to the previous equation, we convert this to exponential form, which gives us 3² = 2x - 5. Simplifying, we get 9 = 2x - 5. Adding 5 to both sides, we have 14 = 2x. Dividing both sides by 2, we find x = 7. Thus, x=4 is not a solution for this equation. The third equation is logx(64) = 4. Converting to exponential form, we get x⁴ = 64. To solve for x, we need to find the fourth root of 64. The fourth root of 64 is 2√2, which is approximately 2.83. Thus, x=4 is not a solution for this equation either. Lastly, we have the equation logx(16) = 4. Converting to exponential form, we get x⁴ = 16. To solve for x, we need to find the fourth root of 16. The fourth root of 16 is 2. Therefore, x=4 is not a solution for this equation. By analyzing each equation, we have identified that the first equation, log₄(3x+4) = 2, is the most likely candidate to have x=4 as a solution. We will now proceed to verify this solution.

Step-by-Step Verification Process

Verifying a potential solution is a critical step in solving equations, especially logarithmic equations. It ensures that the value we found for the variable actually satisfies the original equation and doesn't lead to any undefined terms, such as the logarithm of a negative number or zero. Let's start with the first equation, log₄(3x+4) = 2. We found that x=4 is a potential solution. To verify this, we will substitute x=4 back into the original equation. This gives us log₄(3(4)+4) = 2. Simplifying the expression inside the logarithm, we get log₄(12+4) = 2, which further simplifies to log₄(16) = 2. Now, we need to check if this statement is true. The logarithmic equation log₄(16) = 2 is equivalent to the exponential equation 4² = 16. Since 4² is indeed equal to 16, the equation holds true. Therefore, x=4 is a valid solution for the equation log₄(3x+4) = 2. Now, let's quickly revisit the other equations to confirm that x=4 is not a solution for them. For the equation log₃(2x-5) = 2, we found that x=7 is the solution. Substituting x=4 into this equation gives us log₃(2(4)-5) = 2, which simplifies to log₃(3) = 2. However, log₃(3) = 1, not 2, so x=4 is not a solution. For the equation logx(64) = 4, we found that x is approximately 2.83. Substituting x=4 into this equation gives us log₄(64) = 4. However, log₄(64) = 3, not 4, so x=4 is not a solution. Lastly, for the equation logx(16) = 4, we found that x=2. Substituting x=4 into this equation gives us log₄(16) = 4. However, log₄(16) = 2, not 4, so x=4 is not a solution. This verification process confirms that x=4 is indeed the solution only for the first equation, log₄(3x+4) = 2. The other equations do not have x=4 as a solution.

The Significance of Solution Verification

Verifying solutions in mathematics, particularly in equations involving logarithms, radicals, or rational expressions, is an indispensable practice. It's not merely a procedural step but a crucial safeguard against extraneous solutions. Extraneous solutions are values that emerge during the solving process but do not satisfy the original equation. These can arise due to various algebraic manipulations, such as squaring both sides of an equation or, in the case of logarithms, due to the domain restrictions of logarithmic functions. The domain of a logarithmic function, logb(x), requires that the argument x be strictly positive and the base b be positive and not equal to 1. When solving logarithmic equations, we might obtain values that, when substituted back into the original equation, result in taking the logarithm of a non-positive number, which is undefined. Therefore, verification helps us filter out these extraneous solutions and ensure that we only accept the values that are genuine solutions. In the context of real-world applications, extraneous solutions can lead to incorrect predictions or decisions, making the verification process even more critical. For instance, if we are modeling a physical phenomenon using a logarithmic equation, an extraneous solution might represent a physically impossible scenario. Therefore, by verifying our solutions, we not only ensure mathematical accuracy but also the practical validity of our results. In summary, the significance of solution verification lies in its ability to protect against extraneous solutions, ensure adherence to domain restrictions, and guarantee the accuracy and applicability of the solutions we obtain. It is a fundamental aspect of mathematical problem-solving that should never be overlooked.

Conclusion

In this comprehensive guide, we've explored the process of identifying which equation has x=4 as a solution. We began by understanding the fundamentals of logarithmic equations, emphasizing the relationship between logarithmic and exponential forms. We then systematically analyzed each given equation, converting them to exponential form and solving for x. Through this analysis, we identified log₄(3x+4) = 2 as the most likely candidate. Next, we delved into the crucial step of solution verification, substituting x=4 back into the original equations to confirm its validity. This process not only validated x=4 as a solution for the first equation but also reinforced the importance of verification in preventing extraneous solutions. By working through these equations step-by-step, we have not only found the answer but also deepened our understanding of logarithmic equations and problem-solving strategies in mathematics. Mathematics is not just about finding the right answer; it's about the journey of logical reasoning and critical thinking that leads us there.