Solving Logarithmic Equations $2(log_3 X-1)=log_3 X$ And Finding Y For $y=\sqrt{x}+1$

by Admin 86 views

Logarithmic equations are equations in which the logarithm of an expression containing a variable appears. Solving these equations requires a solid understanding of logarithmic properties and the relationship between logarithms and exponential functions. In this comprehensive guide, we will tackle the equation 2(log⁑3xβˆ’1)=log⁑3x2(\log _3 x-1)=\log _3 x and then explore the connection between its solution and the function y=x+1y=\sqrt{x}+1. Our primary goal is to break down the steps involved in solving logarithmic equations, highlighting the importance of understanding logarithmic properties and demonstrating how these solutions can be used in further mathematical explorations. This article aims to provide a clear and detailed explanation that will be beneficial for students and anyone interested in enhancing their mathematical skills. This exploration will not only solve the given equation but also lay a foundation for tackling more complex problems involving logarithms and square roots.

To solve the logarithmic equation 2(log⁑3xβˆ’1)=log⁑3x2(\log _3 x-1)=\log _3 x, the first step involves simplifying the equation by distributing the 2 on the left side. This gives us 2log⁑3xβˆ’2=log⁑3x2\log _3 x - 2 = \log _3 x. Simplifying logarithmic equations is crucial for isolating the variable and finding the solution. Next, we need to gather all the logarithmic terms on one side of the equation. Subtracting log⁑3x\log _3 x from both sides results in 2log⁑3xβˆ’log⁑3xβˆ’2=02\log _3 x - \log _3 x - 2 = 0, which simplifies to log⁑3xβˆ’2=0\log _3 x - 2 = 0. Now, we isolate the logarithmic term by adding 2 to both sides, giving us log⁑3x=2\log _3 x = 2. At this stage, we can convert the logarithmic equation into its exponential form. Recall that log⁑ba=c\log _b a = c is equivalent to bc=ab^c = a. Applying this to our equation, log⁑3x=2\log _3 x = 2 becomes 32=x3^2 = x. Therefore, x=9x = 9. However, it is crucial to verify the solution in the original equation to ensure that it does not lead to any undefined logarithmic expressions. Substituting x=9x = 9 into the original equation, we get 2(log⁑39βˆ’1)=log⁑392(\log _3 9 - 1) = \log _3 9. Since log⁑39=2\log _3 9 = 2, the equation becomes 2(2βˆ’1)=22(2 - 1) = 2, which simplifies to 2=22 = 2, confirming that x=9x = 9 is a valid solution. This detailed step-by-step approach not only solves the equation but also emphasizes the importance of each step, ensuring a thorough understanding of the process. Furthermore, understanding the properties of logarithms and their interplay with exponential functions is paramount in solving such equations effectively.

Now that we have found the solution to the logarithmic equation, x=9x = 9, we can explore the relationship between this solution and the function y=x+1y = \sqrt{x} + 1. Understanding this relationship involves substituting the value of xx into the function to find the corresponding value of yy. Substituting x=9x = 9 into y=x+1y = \sqrt{x} + 1, we get y=9+1y = \sqrt{9} + 1. Since the square root of 9 is 3, the equation becomes y=3+1y = 3 + 1, which simplifies to y=4y = 4. This calculation reveals that when x=9x = 9, y=4y = 4 for the function y=x+1y = \sqrt{x} + 1. This point (9, 4) represents a coordinate on the graph of the function and highlights the connection between the solution of the logarithmic equation and the values of the given function. Exploring such relationships is fundamental in mathematics as it helps in visualizing solutions and understanding the behavior of different functions. In this context, the function y=x+1y = \sqrt{x} + 1 represents a transformation of the basic square root function, where the graph is shifted upwards by 1 unit. By finding the value of yy corresponding to the solution of the logarithmic equation, we gain a deeper insight into how different mathematical concepts intersect. Moreover, this exercise underscores the significance of substituting solutions into related equations to uncover further insights and relationships. Understanding the properties of functions and their graphical representations is essential for a comprehensive grasp of mathematical principles.

When solving logarithmic equations, it is absolutely essential to verify the solutions obtained. This verification step is crucial because logarithmic functions are only defined for positive arguments. Therefore, any solution that results in taking the logarithm of a non-positive number is extraneous and must be discarded. In the context of our equation 2(log⁑3xβˆ’1)=log⁑3x2(\log _3 x-1)=\log _3 x, we found that x=9x = 9 is a potential solution. To verify this, we substitute x=9x = 9 back into the original equation. As we showed earlier, this substitution leads to a true statement, confirming that x=9x = 9 is indeed a valid solution. However, if we had obtained a different solution, say a negative value or zero, substituting it into the logarithmic equation would have resulted in an undefined expression, indicating that the solution is extraneous. The process of verification helps in avoiding errors and ensures that only correct solutions are considered. This step not only applies to logarithmic equations but also to other types of equations, such as radical equations and rational equations, where extraneous solutions can arise due to the nature of the functions involved. By making verification a routine part of the problem-solving process, mathematicians and students alike can maintain accuracy and confidence in their results. Therefore, always check your solutions in the original equation to guarantee their validity and to ensure that they make sense within the mathematical context of the problem. The rigor of verification is a hallmark of sound mathematical practice.

Logarithmic equations and functions have a wide array of applications in various fields, ranging from science and engineering to finance and computer science. Understanding how to solve logarithmic equations and interpret logarithmic functions is therefore essential for anyone pursuing these disciplines. In science, logarithms are used to describe phenomena that span several orders of magnitude, such as the pH scale in chemistry, which measures the acidity or alkalinity of a solution. Similarly, the Richter scale in seismology uses logarithms to quantify the magnitude of earthquakes. In finance, logarithmic functions are used in calculations involving compound interest and the time value of money. The natural logarithm, in particular, plays a critical role in modeling continuous growth and decay processes. In computer science, logarithms are fundamental in analyzing the efficiency of algorithms. For example, binary search algorithms have a logarithmic time complexity, meaning the time taken to search a sorted list increases logarithmically with the size of the list. This makes binary search highly efficient for large datasets. Furthermore, logarithmic functions are used in information theory and cryptography. The concept of entropy, a measure of uncertainty, is defined using logarithms. Cryptographic algorithms often rely on the properties of logarithms to ensure secure communication. The versatility of logarithmic equations and functions stems from their ability to simplify complex calculations and to model a wide range of real-world phenomena. By mastering the techniques for solving logarithmic equations and understanding the characteristics of logarithmic functions, individuals can unlock powerful tools for problem-solving and analysis in diverse fields. Therefore, the practical applications of logarithms underscore their importance in both theoretical and applied contexts.

When solving logarithmic equations, it is crucial to be aware of common mistakes that can lead to incorrect solutions. One of the most frequent errors is forgetting to verify solutions. As discussed earlier, logarithmic functions are only defined for positive arguments, so it is essential to check whether the solutions obtained satisfy this condition. Another common mistake is incorrectly applying the properties of logarithms. For instance, students sometimes confuse the logarithm of a sum with the sum of logarithms, i.e., assuming that log⁑b(x+y)=log⁑bx+log⁑by\log_b(x + y) = \log_b x + \log_b y, which is incorrect. The correct property is log⁑b(xy)=log⁑bx+log⁑by\log_b(xy) = \log_b x + \log_b y. Similarly, errors can occur when dealing with the change of base formula or when simplifying logarithmic expressions. Another mistake arises from misunderstanding the domain of logarithmic functions. The argument of a logarithm must be positive, and the base must be positive and not equal to 1. Failing to consider these restrictions can lead to extraneous solutions or incorrect interpretations. Additionally, sign errors can occur when manipulating logarithmic equations, especially when distributing coefficients or combining terms. Care must be taken to ensure that all operations are performed correctly. To avoid these mistakes, it is helpful to practice solving a variety of logarithmic equations and to develop a systematic approach to problem-solving. This includes carefully reviewing each step, verifying solutions, and double-checking the application of logarithmic properties. By being mindful of these potential pitfalls and adopting good problem-solving habits, individuals can improve their accuracy and confidence in solving logarithmic equations. Therefore, avoiding these common mistakes is essential for achieving success in this area of mathematics.

In summary, solving the logarithmic equation 2(log⁑3xβˆ’1)=log⁑3x2(\log _3 x-1)=\log _3 x involves several key steps, including simplifying the equation, isolating the logarithmic term, converting to exponential form, and verifying the solution. We found that x=9x = 9 is the valid solution to this equation. Furthermore, we explored the relationship between this solution and the function y=x+1y = \sqrt{x} + 1, demonstrating that when x=9x = 9, y=4y = 4. This exploration highlights the interconnectedness of different mathematical concepts and the importance of understanding how solutions to equations relate to functions. Throughout this guide, we emphasized the significance of verifying solutions in logarithmic equations to avoid extraneous results and the broad applications of logarithms in various fields. By mastering these techniques and principles, students and practitioners can enhance their mathematical skills and problem-solving abilities. The ability to solve logarithmic equations and understand logarithmic functions is a valuable asset in many areas of study and professional endeavors. Therefore, a thorough understanding of logarithmic equations not only helps in solving specific problems but also builds a strong foundation for more advanced mathematical concepts and their applications in the real world. The process of solving equations and exploring their relationships with functions is a fundamental aspect of mathematical thinking and analysis.