Solving 2 Log(x+86) = 4 A Step-by-Step Guide

Jul 9, 2025 by ADMIN 45 views

Solving the Logarithmic Equation: 2 log(x+86) = 4

This article provides a detailed, step-by-step solution to the logarithmic equation 2 log(x+86) = 4. If you're grappling with logarithmic equations or need a refresher on the properties of logarithms, this guide will walk you through the process clearly and concisely. We'll break down each step, explaining the reasoning behind it, so you not only get the answer but also understand the underlying concepts. Whether you're a student preparing for an exam or just curious about solving mathematical problems, this article will equip you with the knowledge and skills you need to tackle similar logarithmic equations with confidence. Let's dive in and unravel the solution together!

Understanding Logarithmic Equations

Before we jump into solving the equation 2 log(x+86) = 4, it's crucial to understand the basics of logarithmic equations. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, if we have an equation like b^y = x, the logarithm (base b) of x is y. This is written as log_b(x) = y. Understanding this fundamental relationship is key to manipulating and solving logarithmic equations. The equation we're dealing with involves a common logarithm, which is a logarithm with base 10. When the base isn't explicitly written, it's assumed to be 10. Therefore, log(x) is the same as log₁₀(x). Knowing this, we can start thinking about how to isolate x in our equation 2 log(x+86) = 4. The properties of logarithms, such as the power rule and the definition of a logarithm, will be our main tools in this process. We'll use these tools to simplify the equation and ultimately find the value of x that satisfies it. Remember, solving logarithmic equations often involves converting them into exponential form, which is where our initial understanding of the relationship between logarithms and exponents becomes crucial.

Step-by-Step Solution

Step 1: Isolate the Logarithm

Our initial equation is 2 log(x+86) = 4. The first step in solving this equation is to isolate the logarithmic term. This means getting the "log(x+86)" part by itself on one side of the equation. To do this, we need to get rid of the coefficient "2" that's multiplying the logarithm. We can accomplish this by dividing both sides of the equation by 2. This is a fundamental algebraic principle – whatever operation we perform on one side of the equation, we must perform on the other side to maintain the equality. Dividing both sides by 2 gives us: (2 log(x+86)) / 2 = 4 / 2. Simplifying this, we get log(x+86) = 2. Now, the logarithmic term is isolated, and we can move on to the next step, which involves converting the equation from logarithmic form to exponential form. This is a crucial step in solving logarithmic equations, as it allows us to eliminate the logarithm and work with a more familiar algebraic structure.

Step 2: Convert to Exponential Form

Now that we have the equation in the form log(x+86) = 2, we need to convert it into exponential form. Remember that the logarithm is base 10 (since no base is explicitly written). The definition of a logarithm tells us that log_b(x) = y is equivalent to b^y = x. Applying this to our equation, where the base b is 10, y is 2, and x is (x+86), we can rewrite the equation as 10² = x + 86. This conversion is the key to unlocking the solution because it removes the logarithm and transforms the equation into a simple algebraic equation that we can easily solve. Understanding the relationship between logarithmic and exponential forms is crucial for solving a wide range of logarithmic problems. By making this conversion, we've taken a significant step towards finding the value of x that satisfies the original equation. Now we have a straightforward equation that we can solve using basic algebraic techniques.

Step 3: Simplify and Solve for x

After converting the equation to exponential form, we have 10² = x + 86. The next step is to simplify the equation and solve for x. First, we simplify 10², which is 10 multiplied by itself, resulting in 100. So, our equation becomes 100 = x + 86. Now, to isolate x, we need to get it by itself on one side of the equation. We can do this by subtracting 86 from both sides of the equation. This maintains the equality and moves us closer to finding the value of x. Subtracting 86 from both sides gives us 100 - 86 = x + 86 - 86. Simplifying this, we get 14 = x. Therefore, the solution to the equation is x = 14. We have now successfully solved the equation by isolating the logarithm, converting to exponential form, and using basic algebraic operations to find the value of x. This step-by-step process demonstrates a clear and effective method for solving logarithmic equations.

Answer

The solution to the equation 2 log(x+86) = 4 is x = 14. Therefore, the correct answer is D. This solution was obtained by first isolating the logarithmic term, then converting the equation to exponential form, and finally solving the resulting algebraic equation for x. This process highlights the importance of understanding the relationship between logarithms and exponents, as well as the fundamental principles of algebraic manipulation. By carefully following these steps, we can confidently solve logarithmic equations and find the correct solutions. It's crucial to remember to always check your answer by plugging it back into the original equation to ensure it satisfies the equation and doesn't result in any undefined operations (like taking the logarithm of a negative number or zero). In this case, substituting x = 14 back into the original equation confirms that it is indeed the correct solution.

Conclusion

In conclusion, we have successfully solved the logarithmic equation 2 log(x+86) = 4 and found the solution to be x = 14. This process involved several key steps, including isolating the logarithm, converting the equation to exponential form, simplifying, and solving for x. Each of these steps relies on a solid understanding of the properties of logarithms and exponents, as well as basic algebraic principles. By mastering these concepts, you can confidently tackle a wide variety of logarithmic equations. Remember, practice is key to improving your problem-solving skills. Work through various examples, paying attention to each step and the reasoning behind it. With practice and a clear understanding of the underlying principles, you'll be able to solve logarithmic equations efficiently and accurately. This problem serves as a good example of how to approach and solve logarithmic equations, providing a solid foundation for further exploration of mathematical concepts.