Solving Logarithmic Equations: A Step-by-Step Guide

Hey everyone, let's dive into solving the logarithmic equation: $\log _6 x+\log _6 3=\log _6(x+1)$. This might seem a bit tricky at first, but trust me, with a few key steps, we can crack it! We're gonna break down the problem, figure out the right way to approach it, and find the value of x that makes everything work. Our goal is to make sure you not only get the right answer but also understand why it's the right answer. Ready to become log-solving pros? Let's get started!

Understanding the Problem: The Core Concepts

Alright, before we jump into the equation, let's make sure we're all on the same page about logarithms. Think of a logarithm as the inverse of exponentiation. When we see $\log _6 x$, it's asking, "To what power must we raise 6 to get x?" The base here is 6. Knowing this helps us to decode the equation, because we are dealing with a sum of logarithms and a single logarithm. This is where those cool logarithm properties come in handy. The most important property for this problem is the product rule for logarithms. This rule tells us that the sum of two logarithms with the same base can be combined into a single logarithm by multiplying the arguments. Remember, guys, understanding the underlying principles makes everything much easier. Let's start with the basics.

The Product Rule of Logarithms

At the heart of solving this equation lies the product rule of logarithms. This rule states that: $\log_b(m) + \log_b(n) = \log_b(m \cdot n)$, where b is the base. It's crucial that the logarithms have the same base, which they do in our equation. This rule allows us to combine two logarithms into one, simplifying the equation considerably. This concept is fundamental for solving this type of problem, so let's make sure we grasp it. It's like having a secret weapon in our mathematical arsenal, giving us the power to transform and solve complex problems easily. Once you have this property down, the rest will be a piece of cake.

Why Logarithms Matter

You might be wondering, "Why are we even bothering with logarithms?" Well, they are incredibly useful in various fields, like calculating compound interest, measuring the intensity of earthquakes (Richter scale), and even in computer science. They simplify calculations involving exponents and are essential in many scientific and engineering applications. Grasping logarithmic equations helps you to build a strong foundation. You are also building important math skills, like problem-solving and critical thinking. They might seem abstract now, but trust me, they pop up in some surprising places later on.

Step-by-Step Solution: Cracking the Code

Now, let's get down to the actual solving of the equation $\log _6 x+\log _6 3=\log _6(x+1)$. Here’s how we're going to do it, step by step, so everyone can follow along. We're going to apply the properties we just discussed to simplify this equation and find that elusive value of x. Let's break it down into manageable chunks. Don’t worry; it's easier than it looks.

Step 1: Combine the Logarithms

First things first, we apply the product rule of logarithms. Remember, $\log _6 x+\log _6 3=\log _6(x \cdot 3)$, which simplifies to $\log _6 (3x)$. So, our equation becomes: $\log _6 (3x) = \log _6 (x+1)$. This is a crucial step! We've taken two terms and turned them into one. Notice how the product rule simplifies the left side. This is like magic, right?

Step 2: Eliminate the Logarithms

Since we have a logarithm on both sides with the same base, we can say that the arguments must be equal. Therefore, $3x = x + 1$. This is the turning point, guys! We've eliminated the logarithms and now have a simple, linear equation to solve. We're getting closer to the solution. The transformation from a logarithmic equation to a linear equation is a critical step in making the problem more manageable. See, we are making progress!

Step 3: Solve for x

Now, we need to isolate x. Subtract x from both sides of the equation: $3x - x = x + 1 - x$, which simplifies to $2x = 1$. Next, divide both sides by 2: $x = \frac{1}{2}$. There you have it! We've found a potential solution for x. But are we done? Not quite!

Step 4: Verification (Important!) – Checking Your Solution

Whenever you solve a logarithmic equation, always check your answer. Plug $x = \frac{1}{2}$ back into the original equation: $\log _6 (\frac{1}{2}) + \log _6 3 = \log _6 (\frac{1}{2} + 1)$. This simplifies to $\log _6 (\frac{1}{2} \cdot 3) = \log _6 (\frac{3}{2})$, which is $\log _6 (\frac{3}{2}) = \log _6 (\frac{3}{2})$. This is true! Therefore, $x = \frac{1}{2}$ is our valid solution. Verification is the critical last step because we need to make sure our solution makes sense in the context of the original problem.

Detailed Explanation of Each Answer Choice

Let’s go through each answer choice to ensure we understand why only one is correct and why the others are incorrect. This helps reinforce the concepts and prepares you for similar problems in the future. We'll look at each option one by one, explaining the logic behind it, and why it is, or isn't, the right answer.

A. $x = \frac{1}{2}$ (Correct)

As we’ve shown, substituting $x = \frac{1}{2}$ into the original equation results in a true statement. This is because $\log _6 (\frac{1}{2}) + \log _6 3$ simplifies to $\log _6 (\frac{3}{2})$, which matches $\log _6 (\frac{1}{2} + 1)$. So, $x = \frac{1}{2}$ satisfies the original equation, making it the correct answer.

B. $x = \frac{3}{2}$ (Incorrect)

If we substitute $x = \frac{3}{2}$ back into the equation, we get $\log _6 (\frac{3}{2}) + \log _6 3 = \log _6 (\frac{3}{2} + 1)$. This simplifies to $\log _6 (\frac{9}{2}) = \log _6 (\frac{5}{2})$. This statement is not true, therefore $x = \frac{3}{2}$ is not a valid solution.

C. $x = 1$ (Incorrect)

Substituting x = 1 into the original equation gives us $\log _6 1 + \log _6 3 = \log _6 (1 + 1)$, which simplifies to $\log _6 3 = \log _6 2$. This statement is also not true. Thus, x = 1 is not a solution to our equation. This reinforces the importance of substitution to verify your answers.

D. $x = \frac{1}{3}$ (Incorrect)

If we substitute $x = \frac{1}{3}$, we get $\log _6 (\frac{1}{3}) + \log _6 3 = \log _6 (\frac{1}{3} + 1)$, which simplifies to $\log _6 1 = \log _6 (\frac{4}{3})$. Because $\log _6 1$ is 0, this means that $0 = \log _6 (\frac{4}{3})$. Hence, the value is also not a solution.

Conclusion: You Got This!

So there you have it! The solution to the equation $\log _6 x+\log _6 3=\log _6(x+1)$ is $x = \frac{1}{2}$. Remember the key steps: use the product rule to combine logarithms, solve the resulting equation, and always check your answer. Solving logarithmic equations is all about understanding the rules and applying them step by step. Keep practicing, and you'll become a pro in no time! Keep up the great work, everyone.