Integers In Interval (-7 Log_3 27) Solution Explained

How many integers are in the interval (-7; log_3 27)? Possible answers: 1) 7 2) 9 3) 10

This article delves into the realm of mathematical problem-solving, specifically focusing on determining the count of integers within a given interval. The problem at hand involves the interval (-7; log_3 27), and our objective is to identify all the whole numbers that lie within this range. This exercise not only reinforces our understanding of number systems but also challenges our ability to manipulate logarithmic expressions. By breaking down the problem into smaller, manageable steps, we can systematically arrive at the solution. We will begin by evaluating the logarithmic expression, then identifying the integers within the defined interval, and finally, counting those integers to provide the final answer. This process highlights the importance of precision and attention to detail in mathematical calculations.

The problem presented to us requires a blend of numerical understanding and logarithmic manipulation skills. At its core, we are asked to determine the cardinality of a set of integers bounded by specific limits. One of these limits is a straightforward negative integer, while the other is expressed as a logarithmic function. The crux of the problem lies in our ability to simplify the logarithmic term and then accurately enumerate the integers that fall within the established range. To embark on this mathematical journey, we will first focus on demystifying the logarithmic component, log_3 27. By converting this expression into a more comprehensible numerical value, we can then proceed to define the interval clearly. This sets the stage for the subsequent step, which involves identifying and counting the integers nestled within the calculated interval. The meticulous execution of each step ensures that we arrive at the correct solution, demonstrating a comprehensive grasp of the underlying mathematical principles.

To begin, let's tackle the logarithmic component of the interval, which is log_3 27. The expression log_3 27 asks the question: "To what power must we raise 3 to obtain 27?" In other words, we are looking for the exponent that satisfies the equation 3^x = 27. Recognizing that 27 is a power of 3, specifically 3^3, we can directly determine that x = 3. Therefore, log_3 27 = 3.

Understanding the logarithmic expression is pivotal for solving the problem. The logarithmic function, in general, is the inverse operation to exponentiation. The expression log_b a = x can be interpreted as "b raised to the power of x equals a." In our case, log_3 27 seeks the power to which 3 must be raised to equal 27. By recognizing that 27 is 3 cubed (3^3), we can simplify the logarithmic expression to its numerical equivalent, which is 3. This simplification transforms the upper bound of our interval from a logarithmic expression to a simple integer, making it easier to identify the integers within the interval. This step is crucial for clarity and precision in the subsequent stages of solving the problem. The accurate evaluation of the logarithmic term lays the foundation for correctly defining the interval and ultimately counting the integers within it. This meticulous approach exemplifies a methodical problem-solving strategy in mathematics.

Now that we've evaluated log_3 27 as 3, our interval becomes (-7; 3). This notation signifies the set of all real numbers greater than -7 and less than 3. Note that the parentheses indicate that the endpoints, -7 and 3, are not included in the interval. Therefore, we are looking for integers strictly between -7 and 3.

Precisely defining the interval is a critical step in solving this problem. The notation (-7; 3) represents an open interval, meaning that the endpoints, -7 and 3, are excluded from the set of numbers under consideration. This distinction is crucial because it affects which integers are included in our final count. The interval encompasses all real numbers greater than -7 and less than 3. To effectively identify the integers within this range, it is helpful to visualize a number line. On this line, we can clearly see the integers that fall between -7 and 3, excluding the endpoints themselves. The clarity in defining the interval ensures that we do not inadvertently include or exclude any integers, leading to an accurate determination of the count of integers. This attention to detail is a hallmark of sound mathematical reasoning and precise problem-solving.

The integers within the interval (-7; 3) are the whole numbers that are greater than -7 and less than 3. Listing them out, we have: -6, -5, -4, -3, -2, -1, 0, 1, and 2. These are the integers that satisfy the condition of being strictly between -7 and 3.

The identification of integers within the defined interval is a fundamental step in solving the problem. Integers are whole numbers (without any fractional or decimal parts), which can be positive, negative, or zero. In the context of our interval (-7; 3), we are seeking all the integers that lie strictly between -7 and 3. This means we need to consider integers greater than -7 but less than 3. By systematically listing these integers, we create a clear set of numbers that we can then count. The accuracy in identifying these integers is paramount, as any omission or inclusion of an incorrect number will affect the final result. The list, comprising -6, -5, -4, -3, -2, -1, 0, 1, and 2, forms the basis for our final calculation. This meticulous approach ensures that we have a complete and accurate set of integers to work with.

By simply counting the integers we've listed, we find that there are 9 integers in the interval (-7; 3): -6, -5, -4, -3, -2, -1, 0, 1, and 2. Thus, the answer to the question is 9.

The final step in solving the problem is to count the integers that we have identified within the interval. This is a straightforward process of enumeration, where we systematically count each integer in our list. The list, comprising -6, -5, -4, -3, -2, -1, 0, 1, and 2, provides a clear and concise set of numbers to work with. By carefully counting each element, we arrive at the total number of integers within the interval. In this case, there are precisely 9 integers that meet the criteria of being greater than -7 and less than 3. This final count provides the definitive answer to the problem, demonstrating our ability to accurately identify and enumerate integers within a specified range. The clarity and precision in this step reflect a comprehensive understanding of the problem and a methodical approach to its solution.

In conclusion, to determine the number of integers in the interval (-7; log_3 27), we first evaluated the logarithmic expression to find that log_3 27 equals 3. This simplified the interval to (-7; 3). Then, we identified all the integers within this interval: -6, -5, -4, -3, -2, -1, 0, 1, and 2. Finally, by counting these integers, we found that there are 9 integers in the given interval. Therefore, the correct answer is 2) 9.

The journey to solve this mathematical problem has highlighted the importance of several key skills. Firstly, the ability to simplify logarithmic expressions is crucial for converting the problem into a more manageable form. Secondly, a clear understanding of interval notation is essential for correctly defining the range of numbers under consideration. Thirdly, the methodical identification and enumeration of integers within the interval are critical for arriving at the accurate solution. By breaking down the problem into smaller, logical steps, we have demonstrated a systematic approach to problem-solving. This approach not only leads to the correct answer but also enhances our comprehension of the underlying mathematical concepts. The successful resolution of this problem reinforces the value of precision, attention to detail, and a step-by-step methodology in mathematical endeavors. The ability to apply these skills is fundamental to tackling more complex mathematical challenges in the future.