Solving 9 Log4(5x - 10) - 6 < 3 A Step-by-Step Guide

In this article, we will delve into the step-by-step process of solving the inequality 9 log₄(5x - 10) - 6 < 3. This problem combines logarithmic functions and inequalities, requiring a solid understanding of both concepts to arrive at the correct solution. Our discussion will not only focus on the mathematical steps but also on the underlying principles that govern logarithmic functions and inequalities. By providing a detailed explanation, we aim to make this guide accessible to anyone with a basic understanding of algebra and logarithms, ensuring that you grasp the method and the reasoning behind each step.

Understanding Logarithmic Inequalities

Before we dive into the specific problem, let's first understand the key concepts surrounding logarithmic inequalities. A logarithmic inequality is an inequality that involves logarithmic functions. Solving these inequalities requires careful consideration of the properties of logarithms, particularly the domain of logarithmic functions and the behavior of logarithmic functions as their arguments change. The domain of a logarithmic function logₐ(x) is restricted to x > 0, as logarithms are only defined for positive arguments. Furthermore, the base of the logarithm, denoted as 'a', must be a positive number not equal to 1. These restrictions are crucial when solving logarithmic inequalities because they determine the valid range of solutions. For instance, in the given inequality 9 log₄(5x - 10) - 6 < 3, the argument of the logarithm, (5x - 10), must be greater than zero. This initial constraint will significantly influence the solution set.

Logarithmic functions are monotonic, meaning they are either strictly increasing or strictly decreasing over their domain. If the base 'a' is greater than 1, the logarithmic function is increasing; if 'a' is between 0 and 1, the function is decreasing. This monotonicity affects how we manipulate inequalities. When the base is greater than 1, the direction of the inequality remains the same when we exponentiate both sides. However, if the base is between 0 and 1, the direction of the inequality must be reversed. Understanding these properties is essential for accurately solving logarithmic inequalities and avoiding common pitfalls.

Step-by-Step Solution

Now, let's tackle the inequality 9 log₄(5x - 10) - 6 < 3 step by step. We will break down each step with detailed explanations to ensure clarity and understanding.

Step 1: Isolate the Logarithmic Term

The first step in solving any inequality involving a logarithmic term is to isolate the term. This involves performing algebraic manipulations to get the logarithmic expression by itself on one side of the inequality. In our case, the inequality is 9 log₄(5x - 10) - 6 < 3. To isolate the logarithmic term, we first add 6 to both sides of the inequality:

9 log₄(5x - 10) - 6 + 6 < 3 + 6

This simplifies to:

9 log₄(5x - 10) < 9

Next, we divide both sides by 9 to further isolate the logarithmic term:

(9 log₄(5x - 10)) / 9 < 9 / 9

Which simplifies to:

log₄(5x - 10) < 1

Step 2: Convert the Inequality to Exponential Form

Once the logarithmic term is isolated, the next step is to convert the inequality from logarithmic form to exponential form. This involves using the definition of a logarithm, which states that logₐ(b) = c is equivalent to aᶜ = b. In our case, we have log₄(5x - 10) < 1. Converting this to exponential form gives us:

4¹ > 5x - 10

This step is crucial as it eliminates the logarithm, allowing us to work with a simpler algebraic inequality. The base of the logarithm, 4, is raised to the power of the value on the other side of the inequality, 1, and this result is set greater than the argument of the logarithm, (5x - 10). Since the base 4 is greater than 1, the direction of the inequality remains the same.

Step 3: Solve the Resulting Inequality

Now that we have converted the logarithmic inequality to an exponential one, we can solve the resulting algebraic inequality. We have:

4 > 5x - 10

To solve for x, we first add 10 to both sides:

4 + 10 > 5x - 10 + 10

Which simplifies to:

14 > 5x

Next, we divide both sides by 5:

14 / 5 > 5x / 5

This gives us:

x < 14/5

So, one part of our solution is x < 14/5.

Step 4: Consider the Domain of the Logarithm

An essential step in solving logarithmic inequalities is to consider the domain of the logarithmic function. The argument of the logarithm must be greater than zero. In our original inequality, the argument is (5x - 10), so we must have:

5x - 10 > 0

Solving this inequality for x, we first add 10 to both sides:

5x - 10 + 10 > 0 + 10

This simplifies to:

5x > 10

Then, we divide both sides by 5:

5x / 5 > 10 / 5

Which gives us:

x > 2

This condition, x > 2, is critical because it restricts the possible values of x to those that make the argument of the logarithm positive. Failing to consider this domain restriction can lead to incorrect solutions.

Step 5: Combine the Inequalities

The final step is to combine the solutions obtained from the algebraic inequality and the domain restriction. We found that x < 14/5 and x > 2. To find the solution set, we need to determine the values of x that satisfy both inequalities. 14/5 is equal to 2.8, so we have x < 2.8 and x > 2. This means that x must be greater than 2 and less than 2.8.

In interval notation, the solution set is (2, 14/5). This interval represents all real numbers between 2 and 2.8, excluding the endpoints. Graphically, this can be represented on a number line with an open circle at 2 and an open circle at 2.8, with the line segment between them shaded.

Common Mistakes to Avoid

When solving logarithmic inequalities, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct solution.

Mistake 1: Ignoring the Domain

One of the most frequent mistakes is neglecting to consider the domain of the logarithmic function. As we discussed earlier, the argument of the logarithm must be greater than zero. Failing to check this condition can lead to including extraneous solutions in the solution set. Always remember to set the argument of the logarithm greater than zero and solve for x before finalizing your solution.

Mistake 2: Incorrectly Applying Logarithmic Properties

Another common mistake is misapplying logarithmic properties. Logarithmic properties, such as the product rule, quotient rule, and power rule, are essential tools for simplifying logarithmic expressions. However, using them incorrectly can lead to errors. For example, students might incorrectly distribute a logarithm over a sum or difference. Always double-check your application of logarithmic properties to ensure accuracy.

Mistake 3: Forgetting to Reverse the Inequality Sign

When dealing with logarithmic functions with a base between 0 and 1, it's crucial to remember to reverse the inequality sign when converting from logarithmic form to exponential form. This is because logarithmic functions with a base between 0 and 1 are decreasing. Forgetting to reverse the sign will result in an incorrect solution. In our problem, the base was 4, which is greater than 1, so we didn't need to reverse the sign. However, if the base were, say, 1/4, we would have had to reverse the sign.

Mistake 4: Algebraic Errors

Simple algebraic errors, such as incorrect addition, subtraction, multiplication, or division, can also lead to incorrect solutions. These errors can occur at any stage of the problem-solving process. To minimize these mistakes, it's helpful to write out each step clearly and double-check your work. Practice and attention to detail are key to avoiding algebraic errors.

Conclusion

Solving logarithmic inequalities requires a thorough understanding of logarithmic properties, algebraic manipulation skills, and careful consideration of the domain of logarithmic functions. By following the step-by-step method outlined in this guide, you can confidently tackle these problems. Remember to isolate the logarithmic term, convert to exponential form, solve the resulting inequality, and, most importantly, consider the domain of the logarithm. By avoiding common mistakes and practicing regularly, you can master the art of solving logarithmic inequalities. The solution to the inequality 9 log₄(5x - 10) - 6 < 3 is the interval (2, 14/5), which represents all real numbers greater than 2 and less than 2.8.