Interval Of Greatest Increase For F(x) = 6 Log₂x - 3
#title: Analyzing Intervals of Greatest Increase for Logarithmic Functions
In this article, we will delve into the concept of the rate of increase of logarithmic functions. Logarithmic functions, unlike linear functions, do not have a constant rate of change. Their rate of change varies depending on the input value, and understanding this variation is crucial in various mathematical and real-world applications. We will specifically analyze the function $f(x) = 6 \log_2 x - 3$ and determine the interval over which it increases at the greatest rate. This analysis will involve understanding the properties of logarithms, derivatives, and how they relate to the rate of change of a function. To fully grasp the solution, we need to understand that the rate of increase of a function is represented by its derivative. The derivative of a logarithmic function decreases as x increases, implying that the function increases at a higher rate for smaller values of x. Therefore, when presented with interval choices, we should look for the interval with the smallest starting value.
Understanding the Function
The function we are examining is $f(x) = 6 \log_2 x - 3$. This is a logarithmic function with base 2, scaled by a factor of 6, and shifted down by 3 units. Logarithmic functions are the inverse of exponential functions, and they exhibit a unique growth pattern. They increase rapidly for small values of x but their rate of increase slows down as x gets larger. This characteristic is fundamental to understanding where the function $f(x)$ increases most rapidly. The constant 6 in front of the logarithmic term stretches the function vertically, amplifying its rate of change, while the -3 simply shifts the entire graph downwards without affecting its rate of change. Thus, the key to finding the interval of greatest increase lies in the logarithmic part, $ \log_2 x $. Understanding the behavior of this component as x varies is crucial. To truly understand this behavior, we can consider the graph of the base logarithmic function, $y = \log_2 x$. This graph starts from negative infinity as x approaches 0, increases rapidly for small positive values of x, and then gradually flattens out as x increases. This flattening out represents the decreasing rate of increase. The scaling factor of 6 in our function magnifies this effect, making the initial rapid increase even more pronounced, and the subsequent slowing down more gradual. This is why the function's rate of change is not constant but rather dependent on the value of x. In practical terms, this behavior is often observed in scenarios involving exponential decay or growth, where the initial changes are the most significant, and the subsequent changes become progressively smaller.
Rate of Change and Derivatives
The rate of change of a function is mathematically represented by its derivative. The derivative, denoted as $f'(x)$, gives the instantaneous rate of change of the function at a particular point x. For logarithmic functions, the derivative provides valuable insights into how the rate of increase varies across different intervals. To find the derivative of our function, $f(x) = 6 \log_2 x - 3$, we first need to recall the derivative of the basic logarithmic function. The derivative of $ \log_b x $ is given by $ rac1}{x \ln b}$, where b is the base of the logarithm and ln represents the natural logarithm. Applying this rule to our function, we get{x \ln 2} = \frac{6}{x \ln 2} $ This derivative function, $f'(x)$, is inversely proportional to x. This means that as x increases, the value of $f'(x)$ decreases, and vice versa. This is a crucial observation because it directly links the rate of increase of $f(x)$ to the value of x. When x is small, $f'(x)$ is large, indicating a high rate of increase. Conversely, when x is large, $f'(x)$ is small, indicating a lower rate of increase. The derivative allows us to not just qualitatively understand the behavior of the function but also to quantitatively compare its rate of increase at different points. For example, if we were to compare the rate of increase at x = 1 and x = 10, the derivative at x = 1 would be significantly larger than the derivative at x = 10, confirming the decreasing rate of increase as x grows. This mathematical precision is what makes derivatives such a powerful tool in analyzing the behavior of functions.
Analyzing the Given Intervals
Now, let's consider the given intervals and analyze where the function $f(x) = 6 \log_2 x - 3$ increases at the greatest rate. The intervals provided are:
A. $$ B. $$ C. $$ D. $$
As we established earlier, the function increases at the greatest rate where its derivative, $f'(x) = \frac{6}{x \ln 2}$, is the largest. Since $f'(x)$ is inversely proportional to x, this occurs at the smallest values of x. Therefore, we need to identify the interval that contains the smallest x values. Comparing the lower bounds of each interval:
- Interval A starts at $ rac{1}{8}$ (0.125)
- Interval B starts at $ rac{1}{2}$ (0.5)
- Interval C starts at 1
- Interval D starts at 2
It is clear that Interval A, $$, has the smallest starting value. This means that the function $f(x)$ will have the highest rate of increase within this interval. The derivative $f'(x)$ will be larger for every x in the interval $$ compared to any x in the other intervals. This is because the denominator in $f'(x)$, which is $x \ln 2$, will be the smallest for values of x in this interval. This logical deduction aligns perfectly with our understanding of the behavior of logarithmic functions and their derivatives. It reinforces the idea that the rate of change is not constant but rather diminishes as x increases. Therefore, when dealing with intervals and seeking the greatest rate of increase, the interval with the smallest x values will invariably be the answer. This approach not only solves the problem efficiently but also deepens our comprehension of the function's dynamics.
Conclusion
In conclusion, the function $f(x) = 6 \log_2 x - 3$ increases at the greatest rate over the interval A. $$. This is because the rate of increase of a logarithmic function is inversely proportional to x, and this interval contains the smallest values of x among the given options. Understanding the relationship between a function, its derivative, and the behavior of logarithmic functions is crucial for solving problems related to rates of change. This analysis highlights the importance of not just memorizing formulas but also grasping the underlying principles that govern mathematical functions. The derivative serves as a powerful tool for understanding and quantifying the rate of change, allowing us to make precise comparisons and deductions. The specific behavior of logarithmic functions, with their decreasing rate of increase as x grows, is a key concept that has broad applications in various fields, from computer science to economics. Therefore, mastering this concept is not just about solving a single problem but also about building a strong foundation for more advanced mathematical and analytical endeavors. By focusing on the fundamental principles, we can develop a deeper understanding and appreciation for the power and elegance of mathematical functions.
