Solving 2ln(2x+3) = 16 A Step-by-Step Guide

When faced with the equation $2 ext{ln}(2x+3) = 16$, finding the solution involves a series of algebraic manipulations and a solid understanding of logarithmic properties. This detailed guide breaks down each step, ensuring a clear and thorough understanding of the solution process. The goal is to isolate $x$ and determine its value, making it crucial to follow each step meticulously. We will use key mathematical principles to unravel the equation, highlighting common techniques used in solving logarithmic equations. By the end of this explanation, you should not only understand the solution but also be capable of tackling similar problems with confidence. This comprehensive approach emphasizes the logical flow of problem-solving, which is essential for success in mathematics. Understanding how each step connects to the overall solution helps in building a strong foundation for more advanced topics. As we delve into the solution, keep in mind the core principle of isolating the variable. This principle guides our every move, from the initial simplification to the final answer. By applying these strategies, we convert a seemingly complex problem into manageable steps, making the solution accessible and understandable.

Step-by-Step Solution

1. Simplify the Equation

Begin by simplifying the equation. We have $2 ext{ln}(2x+3) = 16$. To isolate the natural logarithm, divide both sides of the equation by 2:

ext{ln}(2x+3) = rac{16}{2}

$$ext{ln}(2x+3) = 8$$

This step is crucial because it sets the stage for using the inverse property of logarithms. By dividing both sides by 2, we have reduced the equation to a more manageable form where the natural logarithm is isolated. The natural logarithm, denoted as “ln,” is the logarithm to the base $e$, where $e$ is approximately 2.71828. Understanding this definition is vital for the next steps. Simplifying the equation also helps in visualizing the problem more clearly. The reduced form allows us to focus on the logarithmic part, making it easier to apply the necessary transformations. This initial simplification is a common strategy in solving mathematical problems. It breaks down complexity and allows us to address each part systematically. By reducing the coefficients, we pave the way for applying the inverse function, which will ultimately help us solve for $x$. This foundational step is the cornerstone of our solution process.

2. Convert to Exponential Form

To eliminate the natural logarithm, convert the equation to exponential form. The natural logarithm has a base of $e$, so we rewrite the equation as:

$$e^8 = 2x + 3$$

This step leverages the fundamental relationship between logarithms and exponentials. By understanding that the natural logarithm is the inverse function of the exponential function with base $e$, we can effectively rewrite the equation. This conversion is a pivotal moment in solving the problem, as it transforms the logarithmic equation into a linear equation, which is much easier to handle. The exponential form directly expresses the relationship between the base, exponent, and result, which is critical for solving logarithmic equations. Recognizing this relationship allows us to move from the logarithmic world to the algebraic world, where traditional methods for solving equations apply. When we convert to exponential form, we are essentially undoing the logarithm, making it possible to isolate $x$. This process exemplifies the power of inverse functions in simplifying and solving complex equations. It showcases how understanding these relationships can lead to elegant and efficient solutions. The transition from logarithmic to exponential form is a cornerstone technique in solving logarithmic problems.

3. Isolate the Variable

Now, isolate the variable $x$. Subtract 3 from both sides of the equation:

$$e^8 - 3 = 2x$$

This step is a straightforward application of algebraic principles. By subtracting 3 from both sides, we continue the process of isolating $x$, moving closer to the solution. This manipulation maintains the balance of the equation, a crucial aspect of solving any algebraic problem. Isolating the variable involves performing inverse operations to undo the terms surrounding $x$. Here, subtracting 3 is the inverse operation of adding 3, which brings us closer to our goal. Each step in this isolation process is intentional, aimed at simplifying the equation until we have $x$ on one side and a numerical value on the other. By performing these operations meticulously, we ensure accuracy and avoid potential errors. The strategy of isolating the variable is a fundamental technique in algebra, applicable to a wide range of equation types. This approach allows us to systematically break down the problem, making it easier to find the solution.

4. Solve for x

Finally, divide both sides by 2 to solve for x:

x = rac{e^8 - 3}{2}

This final step completes the isolation of $x$, giving us the solution to the equation. Dividing both sides by 2 effectively undoes the multiplication, revealing the value of $x$. The solution is expressed in terms of $e$, which is appropriate given the original logarithmic equation. The solution x = rac{e^8 - 3}{2} is the exact answer, representing the value that satisfies the original equation $2 ext{ln}(2x+3) = 16$. This methodical approach ensures that we arrive at the correct solution by carefully applying algebraic principles. The solution might seem complex, but it is a direct result of the steps we have taken. Each step has built upon the previous one, leading us to this final expression. This highlights the importance of a systematic approach in solving mathematical problems. The ability to isolate variables and solve equations is a critical skill in mathematics, and this solution demonstrates the power of this skill.

Conclusion

Therefore, the solution to the equation $2 ext{ln}(2x+3) = 16$ is:

x = rac{e^8 - 3}{2}

Thus, the correct answer is:

• A. x = rac{e^8 - 3}{2}

This comprehensive guide has walked you through each step required to solve the logarithmic equation $2 ext{ln}(2x+3) = 16$. We began by simplifying the equation, isolating the natural logarithm, and then converting it to exponential form. From there, we methodically isolated the variable $x$ through algebraic manipulations, eventually arriving at the solution x = rac{e^8 - 3}{2}. Understanding the relationship between logarithmic and exponential forms is crucial in solving these types of equations. The natural logarithm, with its base $e$, is the key to the conversion process, allowing us to transform the equation into a more manageable form. Isolating the variable involves applying inverse operations, such as dividing and subtracting, to gradually strip away the terms surrounding $x$. Each step is deliberate, contributing to the overall goal of finding the value of $x$ that satisfies the original equation. This step-by-step approach not only provides the solution but also enhances understanding of the underlying mathematical principles. By mastering these techniques, you'll be well-equipped to tackle a wide range of logarithmic problems with confidence. The journey from the initial equation to the final solution demonstrates the power of systematic problem-solving in mathematics.

Solving logarithmic equations involves understanding the properties of logarithms and exponentials. In this case, we dealt with a natural logarithm, denoted as $ ext{ln}$, which has a base of $e$. The key concepts include:

• Natural Logarithm: The logarithm to the base $e$ (approximately 2.71828).
• Exponential Form: Converting a logarithmic equation to its equivalent exponential form.
• Inverse Operations: Using inverse operations to isolate the variable.
• Algebraic Manipulation: Applying algebraic principles to solve for $x$.

These keywords are essential for grasping the solution process. Understanding what a natural logarithm is and how it relates to its exponential form is fundamental. Converting between logarithmic and exponential forms is a cornerstone technique in solving these equations. The use of inverse operations, such as division and subtraction, allows us to gradually isolate the variable. Algebraic manipulation involves applying the rules of algebra to rearrange the equation in a way that leads to the solution. By mastering these concepts, you can confidently approach logarithmic equations and solve them efficiently. These keywords also serve as a guide for further exploration and study in mathematics. Each concept is interconnected, forming a network of knowledge that enhances problem-solving skills. The ability to apply these concepts flexibly and creatively is a hallmark of mathematical proficiency.

To solidify your understanding, try solving similar equations such as:

1. $3 ext{ln}(2x+1) = 12$
2. $2 ext{ln}(3x-2) = 8$
3. $5 ext{ln}(x+4) = 15$

Working through these problems will reinforce the steps and concepts discussed. Practice is essential for developing fluency in mathematics. By tackling a variety of problems, you can strengthen your understanding and build confidence in your abilities. Each practice problem is an opportunity to apply the techniques and strategies learned. The more you practice, the more comfortable you will become with the process of solving logarithmic equations. These additional problems also help you identify areas where you may need further review. Solving problems independently is a crucial step in mastering mathematical concepts. It allows you to internalize the methods and develop your problem-solving intuition. Practice transforms theoretical knowledge into practical skills, making you a more proficient mathematician. Engaging with these exercises will undoubtedly enhance your understanding and capabilities.

The solution to the equation $2 ext{ln}(2x+3) = 16$ is x = rac{e^8 - 3}{2}, which corresponds to option A.

This final statement encapsulates the entire solution process and provides the definitive answer. It serves as a concise summary of the journey we have taken, from the initial equation to the final solution. Restating the solution reinforces the key result and ensures clarity. The conclusion highlights the importance of each step in the process, emphasizing the logical flow that led us to the answer. By identifying the correct option, we complete the problem-solving task and confirm our understanding. This conclusive statement leaves the reader with a clear sense of accomplishment and a reinforced understanding of the solution. The conclusion also underscores the importance of accuracy in mathematical problem-solving, ensuring that the final answer is both correct and clearly stated.