Solving The Logarithmic Equation Ln(x² - 2) + Ln(49) = Ln(46)
Hey guys! Today, we're diving into a fun little math problem involving logarithms. We've got this equation: ln(x² - 2) + ln(49) = ln(46), and our mission, should we choose to accept it (and we totally do!), is to find all the solutions for x. Logarithmic equations might seem a bit intimidating at first, but don't worry, we'll break it down step by step and make sure it all makes sense. Think of logarithms as the inverse operation of exponentiation – they help us figure out what exponent we need to raise a base to in order to get a certain number. In this case, we're dealing with the natural logarithm, which has a base of e (that special number approximately equal to 2.71828). So, let's put on our math hats and get started! We'll explore the properties of logarithms, simplify the equation, and ultimately, find the values of x that satisfy the given condition. Buckle up, it's gonna be a logarithmic ride!
Unlocking the Power of Logarithm Properties
Okay, so the first thing we need to do is leverage those cool properties of logarithms. Remember, logarithms have some nifty rules that allow us to manipulate and simplify equations. The key property we'll use here is the product rule of logarithms. This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In mathematical terms, it looks like this: ln(a) + ln(b) = ln(a * b). This rule is a game-changer because it allows us to combine the two logarithmic terms on the left side of our equation into a single logarithm. By applying this property, we can simplify the equation and make it easier to solve. It's like turning two smaller problems into one bigger, but easier, problem! Think of it as merging two streams into a single, more powerful river. So, let's apply this product rule to our equation and see what happens. By combining the logarithmic terms, we're essentially condensing the information and making the equation more manageable. This is a common strategy when dealing with logarithms, and it's a tool you'll find yourself using again and again. Once we've applied the product rule, we'll have a single logarithm on each side of the equation, which will bring us one step closer to isolating x and finding our solutions. Remember, the goal here is to simplify, simplify, simplify! The more we simplify, the easier it will be to solve for the unknown variable.
Applying the Product Rule: A Step-by-Step Guide
Let's get practical and apply the product rule to our equation. We have ln(x² - 2) + ln(49) = ln(46). Using the product rule ln(a) + ln(b) = ln(a * b), we can combine the terms on the left side: ln((x² - 2) * 49) = ln(46). See how we've transformed the sum of two logarithms into a single logarithm of a product? This is the magic of the product rule in action! Now, let's simplify the expression inside the logarithm on the left side. We can distribute the 49 to get: ln(49x² - 98) = ln(46). We're making progress, guys! The equation is looking cleaner and more manageable. This step is crucial because it consolidates the logarithmic terms, setting us up for the next phase of solving for x. By multiplying out the terms, we're essentially unwrapping the expression inside the logarithm, making it easier to work with. Think of it as peeling away the layers of an onion – each step reveals more of the core. Now that we have a single logarithm on each side of the equation, we're in a prime position to get rid of those logarithms altogether. This will involve using the inverse operation of the logarithm, which we'll explore in the next section. So, stay tuned, we're about to take another leap towards finding the solutions for x!
Eliminating Logarithms: The Inverse Operation
Alright, we've successfully applied the product rule and simplified our equation to ln(49x² - 98) = ln(46). Now comes the fun part – getting rid of those pesky logarithms! To do this, we'll use the inverse operation of the natural logarithm, which is exponentiation with the base e. Remember, the natural logarithm asks the question: "To what power must we raise e to get this number?" So, to undo the logarithm, we'll raise e to the power of both sides of the equation. This is based on the fundamental property that if ln(a) = ln(b), then a = b. It's like saying if two logarithmic expressions are equal, then the values inside the logarithms must also be equal. This is a crucial step because it allows us to transition from a logarithmic equation to a more familiar algebraic equation. By exponentiating both sides, we're essentially unwrapping the logarithm and revealing the underlying relationship between the variables. This is a common technique in solving logarithmic equations, and it's a powerful tool to have in your mathematical arsenal. Think of it as using a special key to unlock the equation and reveal its secrets. So, let's take the exponential of both sides and watch the logarithms disappear! We're one step closer to isolating x and finding those elusive solutions.
Taking the Exponential: A Concrete Example
Let's put this into action! We have ln(49x² - 98) = ln(46). To eliminate the natural logarithms, we'll raise e to the power of both sides: e^(ln(49x² - 98)) = e^(ln(46)). Now, here's where the magic happens. Remember that e raised to the power of the natural logarithm of a number simply equals that number. So, on the left side, e^(ln(49x² - 98)) simplifies to 49x² - 98. Similarly, on the right side, e^(ln(46)) simplifies to 46. This is the beauty of inverse operations – they undo each other! Our equation is now transformed into a much simpler algebraic form: 49x² - 98 = 46. Wow, look at how far we've come! We've successfully eliminated the logarithms and are left with a quadratic equation. This is a significant milestone in our problem-solving journey. By taking the exponential of both sides, we've effectively bypassed the logarithmic barrier and entered the realm of standard algebra. This is a common strategy when dealing with logarithmic equations, and it's a testament to the power of inverse operations. Now, we're ready to tackle this quadratic equation and find the values of x that make it true. So, let's move on to the next step: solving for x!
Solving the Quadratic Equation: Finding the Roots
Great job, guys! We've transformed our logarithmic equation into a quadratic equation: 49x² - 98 = 46. Now, it's time to put our algebra skills to work and solve for x. The first step is to get all the terms on one side of the equation, setting it equal to zero. To do this, we'll add 98 to both sides: 49x² = 144. This brings us closer to isolating the x² term. Next, we'll divide both sides by 49 to isolate x²: x² = 144/49. We're almost there! Now, to find x, we need to take the square root of both sides. Remember, when we take the square root, we need to consider both the positive and negative roots. This is a crucial point because quadratic equations often have two solutions. Forgetting the negative root is a common mistake, so let's make sure we don't fall into that trap! By considering both positive and negative roots, we're ensuring that we find all possible values of x that satisfy the equation. Think of it as searching for hidden treasure – you need to explore all possible locations to find it. So, let's take the square root of both sides and see what values of x we uncover. We're on the home stretch now, and the solutions are within our grasp!
Calculating the Square Roots: Unveiling the Solutions
Okay, let's take the square root of both sides of the equation x² = 144/49. The square root of 144 is 12, and the square root of 49 is 7. So, we have two possible solutions: x = ±12/7. That's right, we have two values for x: x = 12/7 and x = -12/7. It's essential to remember the plus or minus sign when taking the square root, as both positive and negative values, when squared, will result in the same positive number. These are our candidate solutions. However, we're not quite done yet! We need to do one crucial thing before we declare victory: check for extraneous solutions. Extraneous solutions are values that we obtain through the algebraic process but don't actually satisfy the original equation. They can arise in logarithmic equations because the logarithm function is only defined for positive arguments. So, we need to make sure that plugging our candidate solutions back into the original equation doesn't result in taking the logarithm of a negative number or zero. This is a critical step in solving logarithmic equations, and it's a way of ensuring that our solutions are valid and meaningful. So, let's put our solutions to the test and see if they hold up!
Checking for Extraneous Solutions: Ensuring Validity
Alright, we've found our candidate solutions: x = 12/7 and x = -12/7. Now, the crucial step: checking for extraneous solutions. Remember, we need to plug these values back into the original equation: ln(x² - 2) + ln(49) = ln(46). The key thing to check is whether the argument of the logarithm, which is x² - 2, is positive for both solutions. If it's not, then that solution is extraneous and we need to discard it. This is because the logarithm function is only defined for positive inputs. Trying to take the logarithm of a negative number or zero results in an undefined value. So, let's start with x = 12/7. We need to calculate (12/7)² - 2. This is where careful arithmetic comes into play! We need to square the fraction and then subtract 2 to see if the result is positive. If it is, then this solution is likely valid. If not, we'll have to reject it. This process of checking for extraneous solutions is like a final quality control step in our problem-solving journey. It ensures that our solutions are not only mathematically correct but also make sense in the context of the original equation. It's a way of catching any errors that might have crept in during the algebraic manipulations. So, let's get calculating and see if our solutions pass the test!
Plugging in the Values: The Moment of Truth
Let's start by checking x = 12/7. We need to evaluate (12/7)² - 2. (12/7)² = 144/49. Now we need to subtract 2, which is the same as subtracting 98/49: 144/49 - 98/49 = 46/49. Since 46/49 is positive, x = 12/7 is a valid solution! Yay! One down, one to go. Now let's check x = -12/7. We need to evaluate ((-12/7)²) - 2. Notice that squaring a negative number gives us a positive number, so ((-12/7)²) = 144/49, which is the same as what we got for the positive value. Therefore, ((-12/7)²) - 2 will also be 46/49, which is positive. So, x = -12/7 is also a valid solution! We've successfully checked both candidate solutions and confirmed that they both satisfy the original equation. This is a great feeling! We've navigated the complexities of logarithmic equations and arrived at the correct answers. So, let's confidently state our final solutions. We've earned it!
Final Solutions: Declaring Victory!
Drumroll, please! We've successfully solved the equation ln(x² - 2) + ln(49) = ln(46), and after checking for extraneous solutions, we've found two valid solutions: x = 12/7 and x = -12/7. These are the values of x that make the equation true. We started with a logarithmic equation, used the properties of logarithms to simplify it, transformed it into a quadratic equation, solved for x, and then rigorously checked our solutions. That's a lot of mathematical ground covered! This journey highlights the power of combining different mathematical concepts and techniques to solve problems. We've used logarithmic properties, algebraic manipulation, and critical thinking to arrive at our final answer. It's like assembling a puzzle – each step builds upon the previous one, leading us to the complete picture. So, let's celebrate our success and remember the valuable skills we've honed along the way. Solving logarithmic equations can be challenging, but with a systematic approach and a solid understanding of the underlying principles, we can conquer any mathematical obstacle that comes our way. Great job, everyone!
Therefore, the solutions to the equation, entered as a comma-separated list, are: 12/7, -12/7.