Hey guys! Today, we are going to dive into solving a logarithmic equation. Specifically, we'll tackle the equation 2log(3x) = 2log(121) and figure out what value of x makes it true. Logarithmic equations might seem intimidating at first, but with a step-by-step approach, they can be quite manageable. So, grab your pencils and let’s get started!

Understanding Logarithmic Equations

Before we jump into solving our particular equation, let's briefly review what logarithmic equations are all about. A logarithm is essentially the inverse operation to exponentiation. In simpler terms, if we have an equation like b^y = x, then we can rewrite it in logarithmic form as log_b(x) = y. Here, b is the base of the logarithm, x is the argument, and y is the exponent. Understanding this relationship is crucial for manipulating and solving logarithmic equations effectively.

Logarithmic equations come in various forms, and the strategies for solving them depend on the specific structure of the equation. Some common techniques include using the properties of logarithms to simplify expressions, converting logarithmic equations to exponential form, and isolating the variable. We'll be using some of these techniques as we solve our equation.

When dealing with logarithms, it's also important to keep in mind certain restrictions. For example, the argument of a logarithm (the value inside the logarithm) must always be positive. This is because logarithms are not defined for non-positive numbers. Similarly, the base of a logarithm must be positive and not equal to 1. These restrictions are essential to consider when checking the validity of solutions to logarithmic equations.

Logarithmic equations are used to solve exponential decay problems, they show up everywhere. They help us calculate things like the magnitude of earthquakes (using the Richter scale), the acidity or alkalinity of a solution (using pH), and even sound intensity (decibels). So, by mastering the art of solving logarithmic equations, you're not just learning a mathematical skill; you're also unlocking the ability to understand and analyze a wide range of real-world phenomena. So stay with me guys!

Step-by-Step Solution

Now, let's get back to our equation: 2log(3x) = 2log(121). Our goal is to isolate x and find its value. Here’s how we can do it:

Step 1: Simplify the Equation

Notice that both sides of the equation have a coefficient of 2 in front of the logarithm. We can simplify the equation by dividing both sides by 2. This gives us:

log(3x) = log(121)

Step 2: Remove the Logarithms

Since we have the same logarithm on both sides of the equation, we can remove the logarithms. This is because if log_b(A) = log_b(B), then A = B. In our case, this means:

3x = 121

Step 3: Isolate x

Now, we need to isolate x by dividing both sides of the equation by 3:

x = 121 / 3

Step 4: Calculate the Value of x

Performing the division, we get:

x ≈ 40.33

So, the value of x that satisfies the equation 2log(3x) = 2log(121) is approximately 40.33. It's always a good idea to plug this value back into the original equation to check if it holds true.

Verification

To verify our solution, let's substitute x = 121/3 back into the original equation:

2log(3 * (121/3)) = 2log(121)

Simplifying the left side, we get:

2log(121) = 2log(121)

Since both sides of the equation are equal, our solution is correct. Therefore, the value of x that satisfies the equation is indeed 121/3 or approximately 40.33. Good job guys!

Alternative Methods

While the method we used above is straightforward, there are other ways to approach this problem. Here’s an alternative method that involves using the properties of logarithms.

Alternative Method 1: Using Logarithmic Properties

Starting with the original equation:

2log(3x) = 2log(121)

We can use the power rule of logarithms, which states that log_b(A^p) = p * log_b(A). Applying this rule in reverse, we can rewrite the equation as:

log((3x)^2) = log(121^2)

Now, we can remove the logarithms:

(3x)^2 = 121^2

Taking the square root of both sides:

3x = 121

Dividing by 3:

x = 121 / 3

As you can see, this method leads us to the same solution as before. It’s always useful to know multiple approaches to solving problems, as different methods may be more efficient in different situations. Keep up the good work!

Common Mistakes to Avoid

When solving logarithmic equations, it’s easy to make mistakes if you’re not careful. Here are some common pitfalls to watch out for:

1. Forgetting to Check for Extraneous Solutions: Always check your solutions by plugging them back into the original equation. Logarithmic equations can sometimes produce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation.
2. Ignoring the Domain of Logarithms: Remember that the argument of a logarithm must be positive. If you find a solution that makes the argument of a logarithm non-positive, that solution is not valid.
3. Misapplying Logarithmic Properties: Make sure you understand and apply the logarithmic properties correctly. Confusing the properties can lead to incorrect solutions.
4. Not verifying the final answer: Always double check if your answer is the correct answer.

By being aware of these common mistakes, you can increase your chances of solving logarithmic equations accurately and efficiently. Remember, practice makes perfect, so keep working on problems to improve your skills. You can do it guys!

Practice Problems

To solidify your understanding of solving logarithmic equations, here are a few practice problems for you to try:

1. Solve for x: log(2x + 5) = log(3x - 2)
2. Solve for x: 2log(x) = log(16)
3. Solve for x: log(x) + log(x - 3) = 1

Try solving these problems on your own, and then check your answers. If you get stuck, review the steps and techniques we discussed earlier. Keep practicing, and you’ll become a pro at solving logarithmic equations in no time!

Conclusion

In this article, we walked through the process of solving the logarithmic equation 2log(3x) = 2log(121). We learned how to simplify the equation, remove the logarithms, isolate the variable, and verify our solution. We also explored an alternative method using logarithmic properties and discussed common mistakes to avoid. By understanding these concepts and practicing regularly, you can confidently tackle a wide range of logarithmic equations. Keep up the great work, and happy problem-solving! Remember guys math is fun!