Hey everyone! Today, we're diving into solving a logarithmic equation. Specifically, we're tackling the equation 2log(3x) = 2log(121). Don't worry, it's not as intimidating as it looks! We'll break it down step by step so you can easily follow along and understand how to find the value of x. Logarithmic equations might seem tricky at first, but with a solid understanding of the basic rules and properties of logarithms, you can solve them with confidence. So, grab your pencils and paper, and let's get started!

Understanding Logarithms

Before we jump right into the equation, let's quickly recap what logarithms actually are. A logarithm is basically the inverse operation to exponentiation. Think of it this way: if we have an equation like b^y = x, then we can rewrite this in logarithmic form as log_b(x) = y. In simpler terms, the logarithm tells you what exponent you need to raise the base (b) to in order to get a certain value (x).

For example, if we have 2^3 = 8, then the logarithmic form would be log_2(8) = 3. This means that we need to raise 2 to the power of 3 to get 8. Make sense?

When you see "log" without a base written (like in our equation), it usually means it's a common logarithm, which has a base of 10. So, log(x) is the same as log_10(x).

Key Properties of Logarithms

To solve logarithmic equations effectively, it's crucial to know some key properties. Here are a few that will come in handy:

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) = p * log_b(m)
4. Equality Rule: If log_b(m) = log_b(n), then m = n (provided b > 0 and b ≠ 1)

These properties allow us to manipulate logarithmic expressions and simplify equations. In our case, we'll be using the equality rule to solve for x.

Solving the Equation 2log(3x) = 2log(121)

Now that we've brushed up on our understanding of logarithms, let's tackle the equation 2log(3x) = 2log(121). Here’s how we can solve it step-by-step:

Step 1: Simplify the Equation

Notice that we have a coefficient of 2 in front of both logarithmic terms. We can simplify the equation by dividing 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 log_b(m) = log_b(n), then m = n. In our case, the base is 10 (since it's a common logarithm), so we can equate the arguments of the logarithms:

3x = 121

Step 3: Solve for x

We now have a simple linear equation. To solve for x, we just need to divide both sides by 3:

x = 121/3

So, the value of x is:

x = 40.333...

Therefore, the solution to the equation 2log(3x) = 2log(121) is x = 121/3, which is approximately 40.333.

Alternative Method: Using the Power Rule

Another way to approach this problem is by using the power rule of logarithms before applying the equality rule. Let's revisit the original equation: 2log(3x) = 2log(121).

Step 1: Apply the Power Rule

We can rewrite the equation using the power rule, which states that p * log_b(m) = log_b(m^p). Applying this rule to both sides of the equation, we get:

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

Step 2: Simplify the Equation

Now we have:

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

Step 3: Apply the Equality Rule

Using the equality rule, we can equate the arguments of the logarithms:

9x^2 = 14641

Step 4: Solve for x

To solve for x, we first divide both sides by 9:

x^2 = 14641/9

x^2 = 1626.777...

Now, take the square root of both sides:

x = ±√(1626.777...)

x ≈ ±40.333

Since the logarithm of a negative number is undefined, we only consider the positive value:

x ≈ 40.333

So, we arrive at the same solution: x = 121/3, which is approximately 40.333.

Common Mistakes to Avoid

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

• Forgetting the Properties: Make sure you have a good grasp of the logarithmic properties. They are essential for simplifying and solving equations correctly.
• Ignoring the Domain: Remember that the argument of a logarithm must be positive. Always check your solutions to make sure they don't result in taking the logarithm of a negative number or zero.
• Incorrectly Applying Rules: Be careful when applying the product, quotient, and power rules. Double-check your work to ensure you're using them correctly.
• Not Simplifying: Always simplify the equation as much as possible before attempting to solve for x. This can make the problem much easier to handle.

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

1. Solve: log(2x + 1) = log(x + 5)
2. Solve: 3log(x) = log(8)
3. Solve: log(x^2 - 4) = log(3x)

Work through these problems step-by-step, and don't hesitate to refer back to the methods we discussed earlier. Practice makes perfect!

Conclusion

So, there you have it! We've successfully solved the equation 2log(3x) = 2log(121) and found that x = 121/3, which is approximately 40.333. Remember the key properties of logarithms and watch out for common mistakes, and you'll be well on your way to mastering logarithmic equations.

Keep practicing, and don't be afraid to ask questions. Happy solving, guys!