Solving Logarithmic Equations: A Step-by-Step Guide

Let's dive into the world of logarithms and tackle the equation: $\ln \sqrt{x+9}=2$. This is a great example to understand how to rewrite logarithmic equations and find their exact solutions. We'll break it down step by step, making sure everything is clear and easy to follow. Get ready to flex those math muscles!

Rewriting the Equation Without Logarithms

Our first mission is to banish the natural logarithm (ln) from the equation. Remember, the natural logarithm, ln, is the logarithm with base e. To get rid of the logarithm, we need to use the definition of a logarithm. If we have an equation in the form $\log_b a = c$, it can be rewritten in exponential form as $b^c = a$.

In our case, the equation is $\ln \sqrt{x+9}=2$. This can be understood as $\log_e \sqrt{x+9}=2$. Using the definition above, we can rewrite the equation in exponential form. Here, b is e, a is $\sqrt{x+9}$, and c is 2. Therefore, the equivalent exponential equation is $e^2 = \sqrt{x+9}$. See, no more logarithms! We have successfully transformed the logarithmic equation into an exponential one. This is a crucial first step because it allows us to isolate x more easily. Remember this fundamental step: always convert logarithms into exponentials to simplify and solve. This step is about understanding the very core of how logarithms and exponentials relate to each other. It's like unlocking the secret code that lets you translate between these two mathematical worlds.

Now we've got $e^2 = \sqrt{x+9}$. The next step is to remove the square root. How do we do that? By squaring both sides of the equation! Squaring both sides of the equation is a legal move because it maintains the equality. Squaring both sides gives us $(e^2)^2 = (\sqrt{x+9})^2$. Simplifying this gives us $e^4 = x+9$. Notice how we progressively get closer to isolating x. Each step simplifies the equation, bringing us closer to the solution. The process is not random; it's a series of strategic moves based on mathematical rules.

This simple rewriting is vital because it transforms a complicated expression into a more manageable form. Always start by identifying the type of logarithm and applying its specific properties. This approach ensures you're on the right track from the beginning. It's like having a map to guide you through a maze. Understanding the underlying properties of logarithms is key to successful problem-solving. It's not just about memorizing formulas; it's about grasping the 'why' behind each step.

Solving the Equation: Finding the Exact Solution

With our equation rewritten as $e^4 = x+9$, we are ready to solve for x. This part is a piece of cake. Our goal is to isolate x on one side of the equation. To do this, we need to get rid of the +9. The solution is straightforward. Subtract 9 from both sides of the equation: $e^4 - 9 = x + 9 - 9$. This simplifies to $x = e^4 - 9$. And there you have it – the exact solution! We have found the value of x that satisfies the original logarithmic equation.

So, what does this solution mean? It means that if we substitute $e^4 - 9$ back into the original equation, it will hold true. Let's briefly check our solution by plugging it back into the initial equation, $\ln \sqrt{x+9}=2$. Substituting $e^4 - 9$ for x, we get $\ln \sqrt{e^4 - 9 + 9} = \ln \sqrt{e^4}$. Knowing that the square root and the square functions are inverse operations, it further simplifies to $\ln e^2$, which, by the nature of the logarithm, simplifies to 2. So the solution checks out perfectly!

The solution set is {e^4 - 9}. Therefore, option A is correct. The other options are incorrect because there is only one exact solution for x that satisfies the original equation. No matter how you try to manipulate the equation, you will always arrive at the same solution. Understanding this process, you will be able to solve a variety of logarithmic equations. The key is to patiently apply the rules of logarithms and algebra until you isolate the variable. Each step is a small victory, and the final solution is the ultimate reward.

Why This Matters

Understanding how to solve logarithmic equations is essential in various fields, including science, engineering, and finance. For instance, in chemistry, logarithmic functions are used to calculate pH values. In finance, they help in calculating compound interest and analyzing investment growth. In physics, logarithms are applied in measuring sound intensity and radioactive decay. This ability to manipulate and solve logarithmic equations opens doors to understanding and solving real-world problems. The skills you acquire here are building blocks for more advanced mathematical concepts and applications. Mastering the basics allows you to confidently tackle more complex problems later on. This is like learning the alphabet before writing a novel.

Solving logarithmic equations is a fundamental skill in mathematics. The process involves converting logarithmic equations into exponential form, simplifying, and isolating the variable. Remember, always double-check your answer by substituting it back into the original equation to ensure its validity. Practice regularly, and you'll find yourself solving these equations with ease. The more you practice, the more intuitive the process becomes. It is like learning a new language: the more you use it, the more natural it feels.

Conclusion

We have successfully navigated the process of rewriting and solving the logarithmic equation $\ln \sqrt{x+9}=2$. We transformed the equation, isolated x, and found the exact solution. This journey has not only solved a mathematical problem but also equipped us with valuable skills. The core takeaway is to understand the relationship between logarithms and exponentials. You can confidently approach any logarithmic equation by breaking it down step by step and using the fundamental properties. Each equation is a new challenge. So embrace it, and enjoy the satisfaction of finding the solution.

To further deepen your understanding, consider exploring more complex logarithmic equations or applications in different fields. Remember, practice is the key to mastering these concepts. Keep exploring, keep learning, and keep growing your mathematical skills. The world of mathematics is vast and fascinating, and every step you take makes you more adept at solving problems and understanding the world around you.

For further reading on logarithms and related topics, you can check out resources like Khan Academy and MathWorld. These sites offer comprehensive explanations, practice problems, and interactive tools to enhance your learning experience.