Solving Logarithmic Equations: A Step-by-Step Guide

Hey guys! Let's dive into solving this logarithmic equation. We'll break it down step-by-step to make sure we understand everything. The problem asks us to find the product of the roots of the equation: $\log_{\frac{1}{3}}(x-3) + 2\log_{\frac{1}{3}}196 = \log_{\frac{1}{3}}x - 2\log_{\frac{1}{3}}13$. Finding the roots of equations can seem tricky at first, but with a solid grasp of logarithmic properties and some careful algebra, we'll crack this one. This guide will walk you through each stage, making sure you grasp every concept. We'll simplify the equation using logarithmic properties, isolate the variable, and then find the values of x that satisfy the equation. This particular problem blends logarithmic rules with algebraic manipulation, so it's a great example to illustrate how different math concepts work together. We'll also need to consider the domain of the logarithmic functions involved, ensuring our solutions are valid. The goal is not just to find the correct answer but also to understand the underlying principles so you can apply them to other problems. So, buckle up, grab your pencils, and let's get started!

Step 1: Combine Logarithms

Our first step is to use the properties of logarithms to simplify the equation. The key here is to combine the logarithmic terms using the properties that govern them. The goal is to get a single logarithm on each side of the equation. This will significantly simplify the process of isolating x and finding the roots. Remember, logarithms are exponents, and their properties are derived from the rules of exponents. We'll use the power rule, which allows us to bring coefficients of logarithms into the exponent of the argument, and the product and quotient rules, which let us combine or separate logarithms with the same base. Let’s begin with the given equation: $\log_{\frac{1}{3}}(x-3) + 2\log_{\frac{1}{3}}196 = \log_{\frac{1}{3}}x - 2\log_{\frac{1}{3}}13$. First, we can apply the power rule of logarithms, where $n \log_b a = \log_b a^n$, to the terms $2\log_{\frac{1}{3}}196$ and $-2\log_{\frac{1}{3}}13$. This transforms our equation. The power rule is crucial here because it deals with the coefficients in front of the logarithms. This is a common strategy when working with logs – changing the equation's form to make it easier to solve. Now the equation becomes: $\log_{\frac{1}{3}}(x-3) + \log_{\frac{1}{3}}196^2 = \log_{\frac{1}{3}}x - \log_{\frac{1}{3}}13^2$. Then we can simplify the squares: $\log_{\frac{1}{3}}(x-3) + \log_{\frac{1}{3}}38416 = \log_{\frac{1}{3}}x - \log_{\frac{1}{3}}169$. By the product rule of logarithms, which states that $\log_b m + \log_b n = \log_b (m \cdot n)$, we can combine the logarithms on the left side: $\log_{\frac{1}{3}}((x-3) \cdot 38416) = \log_{\frac{1}{3}}x - \log_{\frac{1}{3}}169$. On the right side, we can combine the logarithms with the quotient rule, $\log_b m - \log_b n = \log_b (\frac{m}{n})$. Thus, we have: $\log_{\frac{1}{3}}((x-3) \cdot 38416) = \log_{\frac{1}{3}}(\frac{x}{169})$. This simplification is the cornerstone of our solution.

Step 2: Equate Arguments

Now that we have a single logarithm on each side of the equation, both with the same base, we can equate the arguments. The argument of a logarithm is the value inside the logarithm function. This step is valid because the logarithmic function is one-to-one, meaning that if the logarithms of two numbers are equal, then the numbers themselves must be equal. This property allows us to eliminate the logarithms and work with a simpler algebraic equation. So now we have: $(x-3) \cdot 38416 = \frac{x}{169}$. Multiply out the left side, we get $38416x - 115248 = \frac{x}{169}$. The goal here is to isolate x. We are on the path to finding the roots! This is where we transition from logarithmic properties to basic algebra. Cross-multiply to eliminate the fraction: $169(38416x - 115248) = x$. Multiply out to get: $6492214x - 19479432 = x$. Subtract x from both sides: $6492213x - 19479432 = 0$. Now we're getting somewhere! Now add 19479432 to both sides: $6492213x = 19479432$. Finally, isolate x by dividing both sides by 6492213: $x = \frac{19479432}{6492213}$. Calculate this fraction to find the value of x. This value of x that we calculate is our candidate root. But before we get excited, we need to consider something very important... the domain of logarithmic functions.

Step 3: Check Domain and Find Roots

Before we celebrate, we must verify that the solution is within the domain of the original logarithmic equation. Recall that the argument of a logarithm must be positive. This is a fundamental rule, and overlooking it can lead to incorrect or extraneous solutions. Therefore, we must substitute our potential solution back into the original equation to ensure that all logarithmic arguments are positive. The original equation is: $\log_{\frac{1}{3}}(x-3) + 2\log_{\frac{1}{3}}196 = \log_{\frac{1}{3}}x - 2\log_{\frac{1}{3}}13$. First, let's examine the first logarithmic term, $\log_{\frac{1}{3}}(x-3)$. For this to be valid, we must have $x - 3 > 0$, implying $x > 3$. Next, consider the term $\log_{\frac{1}{3}}x$. The argument x must also be positive, so $x > 0$. Lastly, $196$ and $13$ are positive so we don't need to consider that. Combining these restrictions, we find that x must be greater than 3. Thus, our solution must satisfy $x > 3$. Let's calculate the value of x: $x = \frac{19479432}{6492213} = 3$. This is because $19479432 / 6492213 = 3$. Since x is exactly 3, but the domain requires $x > 3$, this solution is invalid. It is an extraneous solution because it does not satisfy the domain restrictions of the original equation. Since there are no valid solutions, the answer is option 4: -. The product of the roots in this case is not applicable since there are no roots.

Step 4: Final Answer

Therefore, after carefully working through the logarithmic equation and considering domain restrictions, we find that the equation $\log_{\frac{1}{3}}(x-3) + 2\log_{\frac{1}{3}}196 = \log_{\frac{1}{3}}x - 2\log_{\frac{1}{3}}13$ has no valid solutions. So, the correct answer is option 4: -, which indicates that there are no roots. We've used key logarithmic properties and a crucial domain check to successfully solve this problem. Understanding these concepts and being able to apply them step by step is crucial for mastering logarithmic equations. Excellent work, guys!