Hey guys! Today, we're diving into a logarithmic equation and figuring out how to solve it step by step. Our mission? To find the value of x in the equation 2log(3x) = 2log(121). Don't worry, it's not as intimidating as it looks! We'll break it down into easy-to-follow steps so you can ace similar problems in the future. So, grab your calculators and let's get started!

Understanding Logarithmic Equations

Before we jump into solving the equation, let's quickly recap what logarithmic equations are all about. A logarithmic equation is an equation that involves logarithms. Remember, logarithms are essentially the inverse of exponential functions. Think of it like this: if you have an exponential equation like b^y = x, you 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.

Now, why are logarithmic equations important? Well, they pop up in various fields like science, engineering, and finance. They're used to model everything from the growth of populations to the decay of radioactive materials. Plus, understanding logarithms can help you make sense of scales that span many orders of magnitude, like the Richter scale for earthquakes or the pH scale for acidity.

When solving logarithmic equations, there are a few key properties to keep in mind:

1. Product Rule: log_b(mn) = log_b(m) + log_b(n)
2. Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
3. Power Rule: log_b(m^p) = plog_b(m)*
4. Equality Rule: If log_b(m) = log_b(n), then m = n

In our case, we'll primarily use the equality rule to solve the equation 2log(3x) = 2log(121). The goal is to isolate x by using these properties and simplifying the equation until we get a clear value for x. So, let's move on to the solution!

Step-by-Step Solution

Okay, let's tackle the equation 2log(3x) = 2log(121) step by step. Follow along, and you'll see how straightforward it is:

Step 1: Simplify the Equation

Notice that we have a '2' multiplying the logarithm on both sides of the equation. To simplify things, we can divide both sides by 2:

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

This simplifies to:

log(3x) = log(121)

Step 2: Apply the Equality Rule

Now that we have log(3x) = log(121), we can use the equality rule. This rule states that if the logarithms are equal, their arguments must also be equal. In other words:

3x = 121

Step 3: Solve for x

To find the value of x, we simply need to divide both sides of the equation by 3:

3x / 3 = 121 / 3

This gives us:

x = 121 / 3

So, the value of x is approximately 40.33. Yay, we solved it!

Alternative Method: Using the Power Rule

Just to show you another way to tackle this problem, let's use the power rule of logarithms. Remember, the power rule states that log_b(m^p) = plog_b(m)*. We can apply this rule in reverse to our original equation:

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

We can rewrite this as:

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

This simplifies to:

log(9x^2) = log(14641)

Now, using the equality rule, we get:

9x^2 = 14641

Divide both sides by 9:

x^2 = 14641 / 9

x^2 = 1626.777...

Taking the square root of both sides:

x = ±√(1626.777...)

x ≈ ±40.33

Since we are dealing with logarithms, we need to check for extraneous solutions. In the original equation 2log(3x) = 2log(121), x must be positive because the argument of a logarithm must be greater than zero. Therefore, we discard the negative solution. So, x ≈ 40.33, which matches our previous result. See? There's often more than one way to crack the code!

Common Mistakes to Avoid

When working with logarithmic equations, it's easy to make a few common mistakes. Here are some pitfalls to watch out for:

1. Forgetting the Domain: Always remember that the argument of a logarithm must be positive. Before you finalize your answer, make sure that the value of x you found doesn't result in taking the logarithm of a negative number or zero.
2. Incorrectly Applying Log Rules: Logarithmic rules can be tricky, so double-check that you're applying them correctly. For instance, log(a + b) is not the same as log(a) + log(b). Similarly, log(a - b) is not the same as log(a) - log(b).
3. Ignoring Extraneous Solutions: Sometimes, when you solve a logarithmic equation, you might end up with solutions that don't actually satisfy the original equation. These are called extraneous solutions. Always plug your solutions back into the original equation to make sure they work.
4. Assuming the Base: If the base of the logarithm isn't explicitly written, it's usually assumed to be 10 (common logarithm). However, be careful not to make this assumption blindly, especially if the problem specifies a different base.
5. Not Simplifying: Always simplify the equation as much as possible before you start solving. This can make the problem much easier to handle. Look for opportunities to combine like terms or use logarithmic properties to reduce the complexity of the equation.

Practice Problems

To really nail down your understanding of 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: log_2(x^2 - 4) = 2
3. Solve for x: log(x) + log(x - 3) = 1
4. Solve for x: 3log(x) = log(8)

Take your time, work through each problem carefully, and don't forget to check your answers. The more you practice, the more comfortable you'll become with solving logarithmic equations. Good luck!

Real-World Applications

Logarithmic equations aren't just abstract mathematical concepts; they're used in a variety of real-world applications. Here are a few examples:

1. Earthquake Measurement: The Richter scale, used to measure the magnitude of earthquakes, is a logarithmic scale. An earthquake with a magnitude of 6 is ten times stronger than an earthquake with a magnitude of 5.
2. Sound Intensity: The decibel scale, used to measure the intensity of sound, is also a logarithmic scale. A sound that is 10 times more intense is 10 decibels louder.
3. Chemistry (pH): The pH scale, used to measure the acidity or alkalinity of a solution, is a logarithmic scale. A solution with a pH of 3 is ten times more acidic than a solution with a pH of 4.
4. Finance: Logarithms are used in finance to calculate compound interest and to model the growth of investments over time.
5. Computer Science: Logarithms are used in computer science to analyze the efficiency of algorithms. For example, the time it takes to search a sorted list using binary search is logarithmic.

Understanding these applications can help you appreciate the practical significance of logarithmic equations and motivate you to learn more about them.

Conclusion

So, there you have it! We've successfully solved the equation 2log(3x) = 2log(121) and found that x = 121/3, or approximately 40.33. We also explored different methods, common mistakes to avoid, practice problems, and real-world applications. Hopefully, this breakdown has made logarithmic equations a little less mysterious and a lot more manageable. Keep practicing, and you'll be solving these equations like a pro in no time!

Remember, mathematics is all about practice and persistence. Don't be afraid to make mistakes—they're part of the learning process. And most importantly, have fun exploring the world of logarithms!