Solving Logarithmic Equations: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into a fun problem involving logarithms. We're going to solve the equation ln(x-7) - ln(3) = ln(2). Don't worry if this looks a bit intimidating at first; we'll break it down step by step and make it super easy to understand. Ready to unlock the secrets of this logarithmic puzzle? Let's get started!

Understanding the Basics of Logarithms

Before we jump into the equation, let's brush up on our logarithm knowledge. Logarithms are the inverse functions of exponentiation. Basically, they help us find the exponent to which a base must be raised to produce a given number. In our equation, we're dealing with the natural logarithm, denoted by ln. The natural logarithm has a base of e (Euler's number, approximately 2.71828). So, when we see ln(x), it's asking, "To what power must we raise e to get x?"

One of the most important properties of logarithms that we'll use is the quotient rule. This rule states that the logarithm of a quotient is equal to the difference of the logarithms. In simpler terms, ln(a) - ln(b) = ln(a/b). This is going to be our secret weapon in solving the given equation. We'll also need to know that if ln(a) = ln(b), then a = b. This seems obvious, but it's crucial for solving the equation. Remember these two properties, and you'll be well on your way to mastering logarithmic equations.

Now, let's talk about the equation itself: ln(x-7) - ln(3) = ln(2). Our goal is to isolate x. To do this, we'll use the properties of logarithms to simplify the equation and get x by itself. We'll combine the terms on the left side, then eliminate the logarithms. Once we do that, we'll have a simple algebraic equation that we can easily solve. This process might seem daunting at first, but trust me, it's pretty straightforward once you get the hang of it. We'll be using the quotient rule to simplify the equation, then using the property that if the logarithms are equal, the arguments are equal too. This way, we'll isolate x and find our solution. Let's get to work!

Step-by-Step Solution of the Logarithmic Equation

Alright, let's get our hands dirty and solve the equation ln(x-7) - ln(3) = ln(2). Follow along closely, and you'll become a logarithm expert in no time! First, we apply the quotient rule of logarithms. As mentioned before, the difference of two logarithms with the same base is equal to the logarithm of the quotient of their arguments. In our case, we have ln(x-7) - ln(3). Using the quotient rule, we combine these two terms into a single logarithm: ln((x-7)/3). So, our equation now looks like this: ln((x-7)/3) = ln(2).

Next, we need to eliminate the logarithms. Since we have ln on both sides of the equation, we can use the property that if ln(a) = ln(b), then a = b. This means we can set the arguments of the logarithms equal to each other: (x-7)/3 = 2. Now, we have a simple algebraic equation that we can easily solve for x. This is the point where the fun really begins! We've transformed a logarithmic equation into something much more manageable. You'll see that it's just a matter of basic algebra to find the value of x.

To solve for x, we need to isolate it. First, let's multiply both sides of the equation by 3 to get rid of the fraction: 3 * ((x-7)/3) = 2 * 3. This simplifies to x - 7 = 6. Now, we just need to add 7 to both sides of the equation to isolate x: x - 7 + 7 = 6 + 7. This gives us our final answer: x = 13. Congratulations, guys! You've solved the logarithmic equation. See? It wasn't as hard as it looked, right? Always remember the properties of logarithms, and you'll conquer these types of equations with ease. Let's move on to verify the solution and make sure our answer is correct.

Verifying the Solution and Potential Pitfalls

Great job solving for x! But wait, we're not quite done yet. We always need to verify our solution to make sure it's correct and that it makes sense in the context of the original equation. Let's plug our solution, x = 13, back into the original equation: ln(x-7) - ln(3) = ln(2). Substitute x with 13: ln(13-7) - ln(3) = ln(2). Simplify the expression inside the first logarithm: ln(6) - ln(3) = ln(2). Now, let's apply the quotient rule again: ln(6/3) = ln(2). Simplify the fraction inside the logarithm: ln(2) = ln(2). Awesome! Our solution checks out. We've verified that our answer is correct.

But wait, there's another crucial step here: checking the domain of the logarithmic function. Logarithms are only defined for positive arguments. So, we need to make sure that when we plug our solution back into the original equation, the arguments of the logarithms are positive. In our original equation, we have ln(x-7) and ln(3). Since 3 is already positive, we're good there. For ln(x-7), we need to ensure that x-7 > 0. Substituting x with 13, we get 13-7 = 6, which is indeed greater than 0. This means our solution is valid and falls within the domain of the logarithmic function. Always remember to check for domain restrictions to avoid making mistakes. Common mistakes include forgetting to check the domain, leading to an incorrect solution. Always verify that your solution doesn't result in taking the logarithm of a negative number or zero. Another common mistake is misapplying the logarithmic properties. Double-check each step to ensure you're using the rules correctly.

Conclusion: Mastering Logarithmic Equations

And there you have it, folks! We've successfully solved the logarithmic equation ln(x-7) - ln(3) = ln(2), and we've verified our answer. We've seen how important it is to understand the properties of logarithms, especially the quotient rule, and how to apply them step by step to solve these kinds of equations. Remember, the key to solving logarithmic equations is to use the properties of logarithms to simplify the equation and isolate the variable. We started with the quotient rule, then simplified the equation, and finally, we made sure to verify our answer and check the domain.

This process may seem complex initially, but with practice, it becomes second nature. Always remember the fundamental rules, such as the quotient rule, and the properties of exponents. Don't be afraid to break down the problem step by step. Write down each step clearly and double-check your work to avoid making mistakes. The more you practice, the more comfortable you'll become with solving logarithmic equations. You'll gain confidence in your ability to tackle even more complex problems. You can explore more examples, practice problems, and detailed explanations of logarithmic properties to deepen your understanding.

So, keep practicing, and don't be discouraged if you get stuck sometimes. Every problem you solve brings you closer to mastering logarithms and other mathematical concepts. Keep exploring, keep learning, and keep having fun with math! You've got this! Now go out there and conquer those logarithmic equations!