Solving 21ogo,s(-x) < Logo,s(7 -6x) A Comprehensive Guide
Delving into the realm of mathematical inequalities, we encounter the intriguing problem of 21ogo,s(-x) < logo,s(7 -6x). This inequality, involving logarithmic functions, presents a fascinating challenge that requires a blend of algebraic manipulation, logarithmic properties, and careful consideration of domain restrictions. In this comprehensive exploration, we will embark on a step-by-step journey to unravel the solution to this inequality, elucidating the underlying concepts and techniques along the way. Our focus will be on providing a clear, concise, and pedagogically sound explanation, suitable for students and enthusiasts alike.
Understanding Logarithmic Functions and Their Properties
Before we plunge into the intricacies of the inequality, it is essential to solidify our understanding of logarithmic functions and their properties. A logarithmic function, denoted as logb(x), where b is the base and x is the argument, is the inverse of an exponential function. In essence, it answers the question: "To what power must we raise the base b to obtain the argument x?" Mathematically, if logb(x) = y, then by = x.
The base b of a logarithmic function plays a crucial role in determining its behavior. Typically, we encounter two common types of logarithmic functions: common logarithms (base 10) and natural logarithms (base e, where e is approximately 2.71828). For the purpose of this discussion, we will assume that the base of the logarithm in the inequality is 10, unless explicitly stated otherwise.
Logarithmic functions possess several key properties that are instrumental in solving equations and inequalities involving them. Some of these properties include:
- Product Rule: logb(xy) = logb(x) + logb(y)
- Quotient Rule: logb(x/y) = logb(x) - logb(y)
- Power Rule: logb(xp) = p logb(x)
- Change of Base Formula: logb(x) = logc(x) / logc(b), where c is any other base
These properties allow us to manipulate logarithmic expressions, simplify equations, and ultimately isolate the variable of interest. In the context of our inequality, these properties will be invaluable in transforming the inequality into a more manageable form.
Addressing Domain Restrictions: A Crucial First Step
Logarithmic functions are not defined for all real numbers. The argument of a logarithmic function must be strictly positive. This fundamental restriction stems from the fact that we cannot raise a positive base to any power and obtain a non-positive result. Therefore, before we proceed with any algebraic manipulations, we must identify the domain restrictions imposed by the logarithmic terms in the inequality.
In our inequality, 21ogo,s(-x) < logo,s(7 -6x), we have two logarithmic terms: log(-x) and log(7 - 6x). To ensure that these terms are well-defined, we must impose the following conditions:
- -x > 0, which implies x < 0
- 7 - 6x > 0, which implies x < 7/6
Combining these two conditions, we find that the domain of the inequality is restricted to x < 0. This means that any solution we obtain must satisfy this condition. Failure to consider domain restrictions can lead to extraneous solutions that do not actually satisfy the original inequality.
Transforming the Inequality: A Journey of Simplification
Now that we have established the domain restrictions, we can proceed with transforming the inequality into a more tractable form. The presence of the coefficient 2 in front of the first logarithmic term suggests that we should utilize the power rule of logarithms to simplify the expression. Applying the power rule, we obtain:
log((-x)2) < log(7 - 6x)
This simplifies to:
log(x2) < log(7 - 6x)
At this point, we have an inequality involving logarithms with the same base. Since the logarithmic function is monotonically increasing for bases greater than 1, we can remove the logarithms without changing the direction of the inequality. This gives us:
x2 < 7 - 6x
We have now transformed the original logarithmic inequality into a quadratic inequality. This is a significant step forward, as we have a wealth of techniques for solving quadratic inequalities.
Solving the Quadratic Inequality: Unveiling the Solution Set
To solve the quadratic inequality x2 < 7 - 6x, we first rearrange it into the standard form:
x2 + 6x - 7 < 0
Next, we factor the quadratic expression:
(x + 7)(x - 1) < 0
To determine the solution set, we analyze the sign of the quadratic expression in different intervals. The roots of the quadratic equation (x + 7)(x - 1) = 0 are x = -7 and x = 1. These roots divide the number line into three intervals: (-∞, -7), (-7, 1), and (1, ∞).
We can now construct a sign table to analyze the sign of the quadratic expression in each interval:
| Interval | x + 7 | x - 1 | (x + 7)(x - 1) |
|---|---|---|---|
| (-∞, -7) | - | - | + |
| (-7, 1) | + | - | - |
| (1, ∞) | + | + | + |
From the sign table, we see that the quadratic expression (x + 7)(x - 1) is negative in the interval (-7, 1). Therefore, the solution to the quadratic inequality x2 + 6x - 7 < 0 is -7 < x < 1.
Considering Domain Restrictions: The Final Refinement
We have obtained the solution to the quadratic inequality, but we must not forget the domain restrictions we identified earlier. The original inequality was defined only for x < 0. Therefore, we must intersect the solution set of the quadratic inequality with the domain restriction.
The intersection of the interval (-7, 1) and the interval (-∞, 0) is the interval (-7, 0). This is the final solution set for the inequality 21ogo,s(-x) < logo,s(7 -6x).
Conclusion: A Triumph of Mathematical Reasoning
In this comprehensive exploration, we have successfully unraveled the solution to the inequality 21ogo,s(-x) < logo,s(7 -6x). Our journey involved a careful blend of algebraic manipulation, logarithmic properties, and consideration of domain restrictions. We transformed the original logarithmic inequality into a quadratic inequality, solved the quadratic inequality, and then refined the solution by considering the domain restrictions.
The final solution set, (-7, 0), represents the set of all real numbers that satisfy the given inequality. This solution exemplifies the power of mathematical reasoning and the importance of adhering to fundamental principles and restrictions. By carefully navigating the intricacies of logarithmic functions and inequalities, we have demonstrated the elegance and effectiveness of mathematical problem-solving.
This exploration serves as a valuable learning experience for students and enthusiasts alike, reinforcing the importance of a solid foundation in mathematical concepts and techniques. By mastering these skills, we can confidently tackle a wide range of mathematical challenges and appreciate the beauty and power of mathematical reasoning.
Logarithmic inequalities, like the one we tackled earlier (21ogo,s(-x) < logo,s(7 -6x)), can often seem daunting at first glance. However, with a systematic approach and a firm grasp of logarithmic properties, these problems can be conquered with confidence. This comprehensive guide will provide you with a step-by-step framework for tackling logarithmic inequalities, empowering you to solve even the most complex problems.
Step 1: Identifying the Domain Restrictions: The Foundation of a Valid Solution
The first and arguably most crucial step in solving any logarithmic inequality is to identify the domain restrictions. As we discussed earlier, logarithmic functions are only defined for positive arguments. Therefore, before you even begin manipulating the inequality, you must ensure that all logarithmic terms are well-defined.
For each logarithmic term in the inequality, set the argument greater than zero and solve for the variable. This will give you a set of inequalities that define the domain of the problem. The solution to the logarithmic inequality must lie within this domain. Failing to consider domain restrictions can lead to extraneous solutions that do not satisfy the original inequality.
In the example of 21ogo,s(-x) < logo,s(7 -6x), we identified the following domain restrictions:
- -x > 0, which implies x < 0
- 7 - 6x > 0, which implies x < 7/6
Combining these conditions, we determined that the domain is restricted to x < 0. This crucial step sets the stage for a valid solution.
Step 2: Simplifying the Inequality: Harnessing the Power of Logarithmic Properties
Once you have established the domain restrictions, the next step is to simplify the inequality using logarithmic properties. These properties, which we discussed earlier, allow you to manipulate logarithmic expressions and transform the inequality into a more manageable form.
Some common techniques for simplifying logarithmic inequalities include:
- Using the Power Rule: logb(xp) = p logb(x) to eliminate coefficients in front of logarithmic terms.
- Using the Product and Quotient Rules: logb(xy) = logb(x) + logb(y) and logb(x/y) = logb(x) - logb(y) to combine logarithmic terms.
- Using the Change of Base Formula: logb(x) = logc(x) / logc(b) to express logarithms in the same base.
By strategically applying these properties, you can often reduce the complexity of the inequality and make it easier to solve. In our example, we used the power rule to transform 21ogo,s(-x) < logo,s(7 -6x) into log(x2) < log(7 - 6x).
Step 3: Removing Logarithms: A Key Transformation
If the inequality involves logarithms with the same base on both sides, you can often remove the logarithms without changing the direction of the inequality (provided the base is greater than 1). This is because the logarithmic function is monotonically increasing for bases greater than 1.
However, it is crucial to remember that this step is only valid if the logarithms have the same base and the base is greater than 1. If the base is between 0 and 1, the logarithmic function is monotonically decreasing, and removing the logarithms will reverse the direction of the inequality.
In our example, since we assumed a base of 10 (which is greater than 1), we could remove the logarithms from log(x2) < log(7 - 6x) to obtain x2 < 7 - 6x.
Step 4: Solving the Resulting Inequality: A Variety of Techniques
After removing the logarithms, you will typically be left with an algebraic inequality, such as a polynomial inequality or a rational inequality. The techniques for solving these inequalities are well-established and depend on the specific form of the inequality.
- Polynomial Inequalities: Rearrange the inequality so that one side is zero, factor the polynomial, find the roots, and use a sign table to determine the intervals where the inequality holds.
- Rational Inequalities: Find the critical points (zeros and points of discontinuity), create a sign table, and determine the intervals where the inequality holds.
In our example, we obtained the quadratic inequality x2 < 7 - 6x. We rearranged it into x2 + 6x - 7 < 0, factored it as (x + 7)(x - 1) < 0, and used a sign table to find the solution -7 < x < 1.
Step 5: Verifying the Solution: The Final Check
Once you have obtained a potential solution set, it is crucial to verify that the solution satisfies both the original inequality and the domain restrictions. This step is essential to eliminate any extraneous solutions that may have arisen during the simplification process.
To verify the solution, you can substitute test values from the solution set into the original inequality and check if the inequality holds. You should also ensure that the test values satisfy the domain restrictions.
In our example, we obtained the solution -7 < x < 1. We then intersected this solution with the domain restriction x < 0 to obtain the final solution -7 < x < 0. This rigorous verification process ensures the accuracy of the solution.
Conclusion: A Path to Mastery
Solving logarithmic inequalities requires a systematic approach and a solid understanding of logarithmic properties and algebraic techniques. By following the five-step framework outlined in this guide, you can confidently tackle even the most challenging logarithmic inequalities.
Remember, the key is to:
- Identify the domain restrictions.
- Simplify the inequality using logarithmic properties.
- Remove logarithms if possible.
- Solve the resulting inequality.
- Verify the solution against the domain restrictions and original inequality.
With practice and perseverance, you can master logarithmic inequalities and expand your mathematical toolkit. Embrace the challenge, and unlock the power of logarithmic problem-solving!
While the five-step framework we discussed provides a solid foundation for solving logarithmic inequalities, some problems may require more advanced techniques and strategies. This section delves into some of these advanced techniques, equipping you with the tools to tackle even the most complex logarithmic inequalities.
1. Dealing with Different Bases: The Change-of-Base Formula to the Rescue
Logarithmic inequalities often involve logarithms with different bases. To effectively compare and manipulate these logarithms, it is essential to express them in the same base. This is where the change-of-base formula comes into play.
The change-of-base formula, which we introduced earlier, states that logb(x) = logc(x) / logc(b), where c is any other base. By strategically choosing a common base (usually 10 or e), you can transform logarithms with different bases into expressions with a single base, making the inequality easier to solve.
For example, consider the inequality log2(x) < log3(x + 1). To solve this inequality, you could use the change-of-base formula to express both logarithms in base 10:
log(x) / log(2) < log(x + 1) / log(3)
Now, you can manipulate the inequality algebraically to isolate x and find the solution set.
2. Substitution: A Powerful Tool for Simplification
In some cases, logarithmic inequalities can become quite complex, with nested logarithmic expressions or repeated logarithmic terms. In such situations, the technique of substitution can be invaluable for simplifying the inequality.
Substitution involves replacing a complex expression with a single variable, making the inequality more manageable. For example, if you encounter an inequality with multiple instances of logb(x), you could substitute y = logb(x) to transform the inequality into a simpler form.
Once you solve for the new variable (y in this case), remember to substitute back to find the solution in terms of the original variable (x).
3. Graphing Techniques: A Visual Approach to Solutions
Graphing can provide a powerful visual aid for solving logarithmic inequalities, especially when analytical methods become cumbersome. By graphing the functions on both sides of the inequality, you can visually identify the intervals where one function is greater than or less than the other.
For example, to solve the inequality log(x) > 1 - x, you could graph the functions y = log(x) and y = 1 - x on the same coordinate plane. The solution to the inequality will be the set of x-values where the graph of y = log(x) lies above the graph of y = 1 - x.
Graphing can also help you visualize the domain restrictions and identify any extraneous solutions.
4. Casework: Dividing and Conquering
Some logarithmic inequalities may require a casework approach, where you divide the problem into different cases based on the values of certain expressions. This is particularly useful when dealing with absolute values or piecewise-defined functions within the logarithmic expressions.
For example, consider the inequality log(|x| + 1) < 1. Since the argument of the logarithm involves an absolute value, we need to consider two cases:
- Case 1: x ≥ 0, in which case |x| = x, and the inequality becomes log(x + 1) < 1.
- Case 2: x < 0, in which case |x| = -x, and the inequality becomes log(-x + 1) < 1.
By solving each case separately and then combining the solutions, you can obtain the complete solution set for the original inequality.
5. The Squeeze Theorem: A Powerful Tool for Bounded Solutions
The squeeze theorem, also known as the sandwich theorem, can be useful for determining the solution to logarithmic inequalities when the variable is bounded between two functions. If you can find two functions, g(x) and h(x), such that g(x) ≤ f(x) ≤ h(x) for all x in an interval, and if the limits of g(x) and h(x) as x approaches a certain value are equal, then the limit of f(x) as x approaches that value is also equal to the same limit.
In the context of logarithmic inequalities, the squeeze theorem can help you determine the behavior of a logarithmic expression when it is bounded between two other functions.
Conclusion: Mastering the Art of Logarithmic Inequalities
Solving complex logarithmic inequalities requires a combination of fundamental principles, advanced techniques, and strategic problem-solving skills. By mastering the techniques discussed in this guide, you can confidently tackle a wide range of logarithmic inequalities and expand your mathematical expertise.
Remember, the key to success is to practice consistently, analyze the problem carefully, and choose the most appropriate techniques for the given situation. With dedication and perseverance, you can unlock the art of logarithmic inequalities and excel in your mathematical pursuits.