Solving Logarithms: Finding The Domain Of 'x'

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Hey guys! Today, we're diving into the world of logarithms. Specifically, we'll tackle a problem where we need to figure out the values of x for which a given logarithm is actually defined. This is super important because logarithms, just like any other mathematical function, have rules. Breaking these rules leads to mathematical nonsense. So, grab your pencils and let's get started! We will be solving the logarithm log_(x^2) sqrt(x^2 - 11x + 18), and determining the values of x for which it is defined. This involves understanding the conditions that must be met for a logarithm to exist.

Understanding the Basics of Logarithms and Their Domains

Alright, before we jump into the problem, let's refresh our memory about what makes a logarithm tick. A logarithm, in its simplest form, looks like this: log_b(a). Here, b is the base and a is the argument. For a logarithm to be defined, both the base and the argument need to play by certain rules. The base (b) needs to be a positive number but not equal to 1. The argument (a) has to be strictly positive. Think of it like this: the base is the foundation, and the argument is what we're actually taking the log of. These rules are non-negotiable! If you break them, you're stepping into the twilight zone of undefined mathematics.

So, applying this to our problem, log_(x^2) sqrt(x^2 - 11x + 18), we have:

  • Base: x^2
  • Argument: sqrt(x^2 - 11x + 18)

Therefore, we've got to ensure two things: x^2 must be positive (and not equal to 1), and sqrt(x^2 - 11x + 18) must be positive. Let's break this down into smaller, more digestible chunks and solve them step by step! This structured approach ensures we don't miss any nuances and that we get the complete set of values for x that make the logarithm valid. We want to be thorough to leave no room for error. The more you practice, the easier it becomes. Let's begin solving.

Condition 1: Analyzing the Base of the Logarithm

Our first hurdle is the base, x^2. The base of a logarithm has to be positive and cannot be equal to 1. So, let's get down to the business of writing this mathematically. We have two conditions regarding the base:

  1. x^2 > 0
  2. x^2 ≠ 1

Let's unpack these conditions one by one, shall we? For x^2 > 0, we can see that x cannot be 0. Any other real number, when squared, gives a positive result. Now, considering x^2 ≠ 1, if we take the square root of both sides, we get x ≠ ±1. Putting it all together, we know that x can be any real number except -1, 0, or 1. This is the first layer of constraints that define the valid values for x. The base of the logarithm dictates a significant portion of the valid solutions. Knowing the conditions is not enough; you must also be able to implement them. The more you familiarize yourself with these conditions, the better you will get!

Condition 2: Analyzing the Argument of the Logarithm

Next, let’s address the argument: sqrt(x^2 - 11x + 18). Remember, the argument of a logarithm must always be positive. This means sqrt(x^2 - 11x + 18) > 0. Since the square root function always returns a non-negative value, we just need the expression inside the square root to be strictly positive (as square roots cannot be negative, and the argument can't be zero either). So, we need to solve the inequality: x^2 - 11x + 18 > 0. This is a quadratic inequality, and we'll solve it by factoring.

First, let's factor the quadratic expression: x^2 - 11x + 18. We're looking for two numbers that multiply to 18 and add up to -11. Those numbers are -2 and -9. Thus, the expression can be factored into (x - 2)(x - 9) > 0. Now we can determine the intervals where this inequality holds true. The roots of the quadratic equation x^2 - 11x + 18 = 0 are x = 2 and x = 9. These are the critical points that divide the number line into intervals. We'll test the intervals to determine where the inequality is satisfied.

  1. Interval 1: x < 2. Let's test x = 0. (0 - 2)(0 - 9) = 18 > 0. So, the inequality is satisfied in this interval.
  2. Interval 2: 2 < x < 9. Let's test x = 5. (5 - 2)(5 - 9) = -12 < 0. So, the inequality is not satisfied in this interval.
  3. Interval 3: x > 9. Let's test x = 10. (10 - 2)(10 - 9) = 8 > 0. So, the inequality is satisfied in this interval.

Therefore, the solution to the inequality x^2 - 11x + 18 > 0 is x < 2 or x > 9. With these intervals in hand, we now know the argument will be positive for these ranges of x. Let's combine the results.

Combining the Conditions to Find the Solution

Alright, we've done some great work so far, folks! We've established two key sets of restrictions on x:

  1. From the base: x cannot be -1, 0, or 1.
  2. From the argument: x < 2 or x > 9.

Now, we need to combine these conditions. This means finding the intersection of these solution sets—the values of x that satisfy both the base and the argument requirements. Remember that the base restrictions, x ≠ -1, x ≠ 0, and x ≠ 1, need to be applied to the solution sets we found for the argument. The base restrictions exclude specific values, while the argument restrictions define intervals where x must lie.

Let’s think this through. The argument condition states x < 2 or x > 9. Now we need to remove any values from these intervals that violate the base conditions. Since x cannot equal -1, 0, or 1, we must exclude these values if they fall within our current intervals.

  • For x < 2, the values -1, 0, and 1 are within this range. So, we adjust our solution: x < -1, -1 < x < 0, 0 < x < 1, or 1 < x < 2.
  • For x > 9, there are no values to exclude from the base's perspective, so x > 9 remains.

Therefore, combining all of this, the solution to the problem is x ∈ (-∞, -1) ∪ (-1, 0) ∪ (0, 1) ∪ (1, 2) ∪ (9, ∞). These are all the values for x that allow our original logarithm to be defined. That means that for any of these values, the base is positive and not equal to 1, and the argument is positive. This makes the logarithm valid and gives it meaning. You're doing great, keep going!

Visualizing the Solution: The Number Line Approach

Sometimes, visualizing the solution on a number line can help solidify your understanding. Let’s map out the restrictions on the number line to see how the solution regions come together. First, mark all the points that are not allowed on the number line: -1, 0, and 1. Then, shade the areas where the argument condition is met: the region to the left of 2 (excluding 2) and the region to the right of 9 (excluding 9).

Finally, we refine the intervals by removing the points -1, 0, and 1. This leaves us with the same solution we determined algebraically: x ∈ (-∞, -1) ∪ (-1, 0) ∪ (0, 1) ∪ (1, 2) ∪ (9, ∞). The number line method is a great tool for understanding inequalities. If you're a visual learner, this technique is a powerful way to organize the data and make it clearer.

Conclusion: The Final Answer

So there you have it, folks! We've successfully navigated the problem of finding the values of x that define the given logarithm, log_(x^2) sqrt(x^2 - 11x + 18). We broke down the problem into smaller, manageable parts, carefully considered the conditions for the base and the argument, and combined these conditions to arrive at the final solution.

The final solution is x ∈ (-∞, -1) ∪ (-1, 0) ∪ (0, 1) ∪ (1, 2) ∪ (9, ∞). This is a comprehensive range of values for which the logarithm is well-defined. Remember, understanding the basic rules of logarithms is essential for solving these types of problems. Without this knowledge, it’s like trying to build a house without a foundation. Understanding the core concepts and applying them step-by-step is key.

Keep practicing, and you'll become a logarithm master in no time! Keep the focus on the conditions, break down the problem step by step, and don’t be afraid to double-check your work. You've got this!

Further Exploration and Practice

To solidify your understanding, here are some suggestions:

  • Practice problems: Try solving similar logarithm problems with different bases and arguments. Change up the numbers and the expressions. The more you work through different scenarios, the more comfortable you'll become.
  • Online resources: Explore online resources like Khan Academy, which offer detailed explanations, video tutorials, and practice exercises. Use these resources to get further explanations of concepts that may be difficult.
  • Review and reinforce: Regularly review the conditions that define logarithms. This includes both the base and the argument rules. Understanding these rules is fundamental to solving any logarithm problem. If you forget these, you are lost!
  • Teach someone else: Teaching the concept to someone else is a great way to reinforce your understanding. Explaining the principles to someone else can highlight gaps in your understanding.

By following these steps, you will continue to increase your understanding of logarithms! Always remember, math is a skill that improves with practice, so keep working at it, guys. Keep the math juices flowing, and happy calculating!