Solve Log₄(2x-3)=2 A Step-by-Step Guide
In the realm of mathematics, logarithmic equations often present a unique challenge, requiring a blend of algebraic manipulation and a solid understanding of logarithmic properties. This comprehensive guide delves into the intricacies of solving the equation log₄(2x-3)=2, providing a step-by-step approach that not only leads to the solution but also illuminates the underlying principles of logarithmic functions. Whether you're a student grappling with logarithmic equations for the first time or a seasoned mathematician seeking a refresher, this article aims to equip you with the knowledge and skills necessary to tackle similar problems with confidence.
Before we dive into the solution, let's first establish a firm grasp of what logarithms are. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, if we have an equation like b^y = x, the logarithm of x to the base b is y. This is written as log_b(x) = y. The base, b, is a crucial component of the logarithm, indicating the number that is being raised to a power. In our given equation, log₄(2x-3)=2, the base is 4.
Logarithms are fundamental tools in various fields, including mathematics, physics, engineering, and computer science. They are used to simplify complex calculations, model exponential growth and decay, and solve equations involving exponents. Understanding the properties of logarithms is essential for effectively manipulating and solving logarithmic equations. Key properties include the product rule (log_b(mn) = log_b(m) + log_b(n)), the quotient rule (log_b(m/n) = log_b(m) - log_b(n)), and the power rule (log_b(m^p) = p*log_b(m)). These properties allow us to rewrite logarithmic expressions in different forms, making them easier to work with.
The importance of logarithms extends beyond theoretical mathematics. In practical applications, logarithms are used to measure the intensity of earthquakes (the Richter scale), the loudness of sound (decibels), and the acidity of solutions (pH scale). They also play a vital role in data analysis and statistical modeling, where logarithmic transformations are often used to normalize data and make it easier to interpret. By mastering the concepts and techniques related to logarithms, you'll not only enhance your mathematical skills but also gain valuable tools for solving real-world problems.
Now, let's tackle the equation log₄(2x-3)=2. The key to solving this equation lies in converting it from logarithmic form to exponential form. Recall that log_b(x) = y is equivalent to b^y = x. Applying this principle to our equation, we can rewrite log₄(2x-3)=2 as 4² = 2x-3. This transformation eliminates the logarithm, making the equation easier to solve.
Our equation now reads 4² = 2x-3. Simplifying the left side, we get 16 = 2x-3. The next step is to isolate the term containing x. We can do this by adding 3 to both sides of the equation, resulting in 16 + 3 = 2x, which simplifies to 19 = 2x. To solve for x, we need to divide both sides of the equation by 2. This gives us x = 19/2, or x = 9.5.
However, our work isn't quite done yet. It's crucial to check our solution in the original equation to ensure it's valid. Logarithms are only defined for positive arguments, so we need to make sure that 2x-3 is greater than zero when x = 9.5. Substituting x = 9.5 into the expression 2x-3, we get 2(9.5) - 3 = 19 - 3 = 16, which is indeed greater than zero. Therefore, our solution x = 9.5 is valid. This step of checking the solution is essential in solving logarithmic equations, as it helps to identify and eliminate extraneous solutions that may arise during the algebraic manipulation.
When solving logarithmic equations, several common mistakes can lead to incorrect solutions. One of the most frequent errors is forgetting to check the solution in the original equation. As we discussed earlier, logarithmic functions are only defined for positive arguments, so it's crucial to ensure that the expression inside the logarithm is positive when you substitute your solution. Failing to do so can result in extraneous solutions that don't actually satisfy the original equation.
Another common mistake is misapplying the properties of logarithms. The product rule, quotient rule, and power rule are powerful tools for manipulating logarithmic expressions, but they must be applied correctly. For example, log_b(m+n) is not equal to log_b(m) + log_b(n). Similarly, log_b(m-n) is not equal to log_b(m) - log_b(n). It's essential to understand the correct application of these rules to avoid errors.
Furthermore, students sometimes struggle with converting between logarithmic and exponential forms. Remember that log_b(x) = y is equivalent to b^y = x. This conversion is the key to solving many logarithmic equations, so it's important to master this skill. Practice converting equations back and forth between these forms to solidify your understanding.
Finally, be mindful of the domain of logarithmic functions. The argument of a logarithm must be positive, and the base must be positive and not equal to 1. Keep these restrictions in mind when solving logarithmic equations, and always check your solutions to ensure they fall within the domain of the logarithmic functions involved.
To solidify your understanding of solving logarithmic equations, let's work through a few practice problems.
- Solve for x: log₂(3x + 1) = 3
- Solve for x: log₅(2x - 5) = 1
- Solve for x: log₃(x²) = 2
For the first problem, log₂(3x + 1) = 3, we can convert the equation to exponential form: 2³ = 3x + 1. This simplifies to 8 = 3x + 1. Subtracting 1 from both sides gives us 7 = 3x, and dividing by 3 yields x = 7/3. We need to check if this solution is valid by substituting it back into the original equation: 3(7/3) + 1 = 7 + 1 = 8, which is positive. Therefore, x = 7/3 is a valid solution.
In the second problem, log₅(2x - 5) = 1, we convert to exponential form: 5¹ = 2x - 5. This simplifies to 5 = 2x - 5. Adding 5 to both sides gives us 10 = 2x, and dividing by 2 yields x = 5. Checking the solution: 2(5) - 5 = 10 - 5 = 5, which is positive. So, x = 5 is a valid solution.
The third problem, log₃(x²) = 2, converts to 3² = x², which simplifies to 9 = x². Taking the square root of both sides gives us x = ±3. We need to check both solutions. For x = 3, we have 3² = 9, which is positive. For x = -3, we also have (-3)² = 9, which is positive. Therefore, both x = 3 and x = -3 are valid solutions.
These practice problems demonstrate the key steps involved in solving logarithmic equations: converting to exponential form, solving the resulting algebraic equation, and checking for extraneous solutions. By working through these examples, you can develop your problem-solving skills and gain confidence in your ability to tackle more complex logarithmic equations.
In this comprehensive guide, we've explored the process of solving the logarithmic equation log₄(2x-3)=2. We've covered the fundamental concepts of logarithms, the step-by-step solution process, common mistakes to avoid, and practice problems to solidify your understanding. By mastering these concepts and techniques, you'll be well-equipped to tackle a wide range of logarithmic equations.
Solving logarithmic equations is a skill that requires a combination of algebraic manipulation and a solid understanding of logarithmic properties. Remember to always convert the equation to exponential form, solve for the variable, and check your solution in the original equation to avoid extraneous solutions. With practice and patience, you can master this essential mathematical skill and apply it to various real-world problems. Logarithmic equations are a crucial part of mathematics, and understanding them opens doors to more advanced topics and applications. Mastering logarithmic equations is a valuable asset in your mathematical journey.
