Solving Logarithmic Equations A Step-by-Step Guide To Log₂(3x - 7) = 3
Introduction
In the realm of mathematics, logarithmic equations play a crucial role in various fields, including calculus, algebra, and applied sciences. Understanding how to solve these equations is fundamental for students, engineers, and anyone dealing with quantitative analysis. This article provides a detailed, step-by-step guide on solving the specific logarithmic equation log₂(3x - 7) = 3, ensuring clarity and comprehension for all readers. Our focus is on demystifying the process, making it accessible, and enhancing your problem-solving skills in mathematics. We'll cover the foundational concepts of logarithms, the properties that govern them, and how these principles apply to finding the solution for this particular equation. Mastering this technique not only helps in solving similar problems but also provides a solid base for tackling more complex mathematical challenges. The goal here is to break down each step, making the solution process as transparent and easy to follow as possible, thereby building your confidence in handling logarithmic equations.
Understanding Logarithms
At its core, a logarithm is the inverse operation to exponentiation. To fully grasp the solution to our equation, log₂(3x - 7) = 3, we must first define what a logarithm represents. In simple terms, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Mathematically, if we have the equation logₐ(b) = c, this means that aᶜ = b. Here, 'a' is the base of the logarithm, 'b' is the argument (the number we're taking the logarithm of), and 'c' is the logarithm itself, which represents the power to which 'a' must be raised to obtain 'b'. Understanding this relationship is crucial for manipulating and solving logarithmic equations. In our specific equation, log₂(3x - 7) = 3, the base is 2, the argument is (3x - 7), and the logarithm is 3. To solve for x, we need to convert this logarithmic form into its equivalent exponential form. This conversion is the key first step in unraveling the equation and isolating the variable x. By understanding the fundamental nature of logarithms and their relationship with exponential functions, we lay a solid groundwork for the subsequent steps in solving our equation. This foundation allows us to approach the problem systematically, ensuring we apply the correct principles and arrive at the accurate solution.
Converting Logarithmic to Exponential Form
The first step in solving the equation log₂(3x - 7) = 3 is to convert it from logarithmic form to exponential form. As we discussed earlier, the logarithmic equation logₐ(b) = c is equivalent to the exponential equation aᶜ = b. Applying this principle to our equation, where a = 2, b = (3x - 7), and c = 3, we can rewrite log₂(3x - 7) = 3 as 2³ = (3x - 7). This transformation is crucial because it eliminates the logarithm, allowing us to work with a more straightforward algebraic equation. By converting to exponential form, we've essentially unwrapped the logarithmic operation, making the variable 'x' more accessible. The equation 2³ = (3x - 7) is now in a form that we can easily manipulate using standard algebraic techniques. This step is a fundamental application of the definition of logarithms and demonstrates the powerful relationship between logarithms and exponents. It's important to remember this conversion process as it's a common technique used in solving various logarithmic equations. Mastering this conversion ensures that you can efficiently tackle a wide range of problems involving logarithms.
Simplifying the Exponential Expression
Now that we have the equation in exponential form, 2³ = (3x - 7), the next step is to simplify the exponential expression. Evaluating 2³ gives us 2 * 2 * 2, which equals 8. So, our equation now becomes 8 = (3x - 7). Simplifying the exponential term is a crucial step as it reduces the complexity of the equation, making it easier to solve for 'x'. This simplification transforms the equation into a simple linear equation, which we can solve using basic algebraic techniques. By calculating the value of 2³, we've moved closer to isolating the variable 'x' and finding its value. This step highlights the importance of understanding basic arithmetic operations and how they apply in the context of solving equations. With the simplified equation 8 = (3x - 7), we are now well-positioned to proceed with the next steps, which involve isolating 'x' by performing algebraic manipulations. This methodical approach ensures accuracy and clarity in the solution process.
Isolating the Variable
With the simplified equation 8 = (3x - 7), our next objective is to isolate the variable 'x'. To do this, we need to perform algebraic operations that will eventually leave 'x' by itself on one side of the equation. The first step in isolating 'x' is to eliminate the constant term on the right side of the equation. We can achieve this by adding 7 to both sides of the equation. This maintains the balance of the equation and helps us move closer to isolating 'x'. Adding 7 to both sides gives us 8 + 7 = 3x - 7 + 7, which simplifies to 15 = 3x. Now, we have a simpler equation where 'x' is multiplied by a coefficient. To completely isolate 'x', we need to eliminate this coefficient. We can do this by dividing both sides of the equation by 3. This gives us 15 / 3 = 3x / 3, which simplifies to 5 = x. Therefore, we have successfully isolated the variable 'x' and found its value. This process of isolating a variable is a fundamental technique in algebra and is essential for solving various types of equations. By carefully applying algebraic operations, we can systematically work towards our goal of finding the value of the unknown variable. In this case, we have found that x = 5, but it's important to verify this solution to ensure it satisfies the original equation.
Verifying the Solution
After finding a potential solution, it is crucial to verify it by substituting the value back into the original equation. This step ensures that our solution is correct and that we haven't made any errors in our calculations. In our case, we found that x = 5. To verify this solution, we substitute x = 5 into the original equation, log₂(3x - 7) = 3. This gives us log₂(3 * 5 - 7) = 3. Now, we simplify the expression inside the logarithm: 3 * 5 = 15, and 15 - 7 = 8. So, our equation becomes log₂(8) = 3. To check if this is true, we need to determine if 2 raised to the power of 3 equals 8. Indeed, 2³ = 2 * 2 * 2 = 8. Therefore, log₂(8) = 3 is a true statement, confirming that our solution x = 5 is correct. Verifying the solution is an essential step in solving any equation, especially logarithmic equations, as it helps to catch any potential errors and ensures accuracy. By substituting the solution back into the original equation, we can confidently conclude that x = 5 is the correct answer to the equation log₂(3x - 7) = 3. This practice reinforces the importance of precision and attention to detail in mathematical problem-solving.
Conclusion
In conclusion, we have successfully navigated the process of solving the logarithmic equation log₂(3x - 7) = 3. We began by understanding the fundamental definition of logarithms and their relationship to exponents. This understanding allowed us to convert the logarithmic equation into its equivalent exponential form, which is a crucial step in solving such equations. We then simplified the exponential expression and proceeded to isolate the variable 'x' using algebraic manipulations. After finding a potential solution, x = 5, we emphasized the importance of verification by substituting this value back into the original equation. This step confirmed the correctness of our solution, reinforcing the need for precision and accuracy in mathematical problem-solving. This comprehensive approach not only solves the specific equation but also provides a framework for tackling other logarithmic equations. By mastering these steps, you can confidently approach and solve a wide range of logarithmic problems. Understanding and applying these principles will undoubtedly enhance your mathematical skills and problem-solving abilities. The journey through this equation highlights the interconnectedness of mathematical concepts and the power of systematic problem-solving.
