Interval Of Maximum Increase For Logarithmic Function F(x)
The question at hand delves into the heart of calculus, specifically focusing on the rate of change of a logarithmic function. To accurately pinpoint the interval where the function $f(x) = 6 \log_2 x - 3$ increases most rapidly, we need to understand the concept of the derivative. The derivative of a function, denoted as $f'(x)$, provides us with the instantaneous rate of change of the function at any given point. In simpler terms, it tells us how much the function's value is changing for a tiny change in the input, $x$. For a function to be increasing, its derivative must be positive, indicating that the function's value is growing as $x$ increases. Furthermore, the magnitude of the derivative gives us the rate of increase; the larger the derivative, the faster the function is increasing.
Now, let's apply this knowledge to our specific function, $f(x) = 6 \log_2 x - 3$. The first step is to find its derivative. Recall that the derivative of $\log_a x$ is given by $\frac{1}{x \ln a}$. Therefore, the derivative of $6 \log_2 x$ is $6 \cdot \frac{1}{x \ln 2}$, and the derivative of the constant term -3 is simply 0. Combining these results, we find the derivative of our function to be:
This derivative, $f'(x)$, is the key to unlocking the mystery of the function's rate of change. By analyzing this expression, we can determine where the function is increasing and, more importantly, where it is increasing at the greatest rate. Observe that $f'(x)$ is always positive for $x > 0$, since both $x$ and $\ln 2$ are positive in this domain. This confirms that the function $f(x)$ is indeed increasing for all positive values of $x$. However, the rate of increase is not constant; it depends on the value of $x$.
To find the interval where the function increases most rapidly, we need to consider how $f'(x)$ changes as $x$ varies. Notice that $f'(x)$ is inversely proportional to $x$. This means that as $x$ increases, $f'(x)$ decreases, and vice versa. In other words, the rate of increase of the function is highest when $x$ is smallest. This is a crucial observation that guides us to the correct answer.
With this understanding, we can now examine the given intervals and determine which one corresponds to the smallest values of $x$. The intervals provided are:
A. $[2, 6]$ B. $[\frac{1}{8}, \frac{1}{2}]$
Comparing these intervals, we see that the interval B, $[\frac{1}{8}, \frac{1}{2}]$, contains smaller values of $x$ than the interval A, $[2, 6]$. Since the rate of increase is highest when $x$ is smallest, we can conclude that the function $f(x)$ increases most rapidly over the interval $[\frac{1}{8}, \frac{1}{2}]$. Therefore, the correct answer is B.
In summary, by understanding the concept of the derivative and analyzing the behavior of $f'(x)$, we successfully identified the interval where the logarithmic function $f(x) = 6 \log_2 x - 3$ increases at the greatest rate. This problem highlights the power of calculus in understanding the dynamics of functions and their rates of change.
Option B: The Interval of Maximum Increase Explained
Delving deeper into why option B, the interval $[\frac{1}{8}, \frac{1}{2}]$, is the correct answer necessitates a more granular examination of the derivative function, $f'(x) = \frac{6}{x \ln 2}$. As established earlier, this derivative is inversely proportional to $x$, meaning that the smaller the value of $x$, the larger the value of $f'(x)$. This inverse relationship is the cornerstone of our understanding.
Consider the interval $[\frac{1}{8}, \frac{1}{2}]$. The values of $x$ within this interval range from 0.125 to 0.5. Let's compare this to the other interval, $[2, 6]$, where $x$ ranges from 2 to 6. It's immediately clear that the values of $x$ in $[\frac{1}{8}, \frac{1}{2}]$ are significantly smaller than those in $[2, 6]$. This difference in magnitude of $x$ directly translates to a difference in the magnitude of the derivative, $f'(x)$. When $x$ is small, the denominator of $f'(x)$ is small, making the overall value of $f'(x)$ large. Conversely, when $x$ is large, the denominator of $f'(x)$ is large, making the overall value of $f'(x)$ small.
To further illustrate this point, let's calculate the value of $f'(x)$ at the endpoints of each interval.
For the interval $[\frac{1}{8}, \frac{1}{2}]$:
- At $x = \frac{1}{8}$, $f'(\frac{1}{8}) = \frac{6}{(\frac{1}{8}) \ln 2} = \frac{48}{\ln 2} \approx 69.25$. This represents a substantial rate of increase.
- At $x = \frac{1}{2}$, $f'(\frac{1}{2}) = \frac{6}{(\frac{1}{2}) \ln 2} = \frac{12}{\ln 2} \approx 17.31$. While smaller than the rate at $x = \frac{1}{8}$, it's still a significant rate of increase.
For the interval $[2, 6]$:
- At $x = 2$, $f'(2) = \frac{6}{2 \ln 2} = \frac{3}{\ln 2} \approx 4.33$. This rate of increase is considerably smaller than those observed in the interval $[\frac{1}{8}, \frac{1}{2}]$.
- At $x = 6$, $f'(6) = \frac{6}{6 \ln 2} = \frac{1}{\ln 2} \approx 1.44$. This represents the smallest rate of increase among all the calculated values.
The numerical examples vividly demonstrate the effect of $x$ on the rate of increase. The smaller the $x$, the steeper the increase in the function's value. Within the interval $[\frac{1}{8}, \frac{1}{2}]$, the function exhibits the highest rate of increase because this interval encompasses the smallest values of $x$ among the given options. Therefore, our analysis solidifies the conclusion that option B is indeed the correct answer.
In essence, selecting the correct interval boils down to understanding the fundamental relationship between the input variable $x$ and the rate of change, as dictated by the derivative of the function. The inverse proportionality between $x$ and $f'(x)$ is the key to unlocking this problem and appreciating the behavior of logarithmic functions.
The Significance of Logarithmic Function Derivatives
The problem we've dissected isn't just a mathematical exercise; it's a gateway to understanding the broader significance of derivatives in analyzing logarithmic functions. Logarithmic functions, with their unique properties, play a pivotal role in various fields, from physics and engineering to economics and computer science. Understanding their rates of change is crucial for modeling and predicting real-world phenomena.
Derivatives provide a powerful lens through which we can examine the dynamic behavior of these functions. They allow us to quantify the instantaneous rate of change, revealing how the function's output responds to infinitesimal changes in its input. This is particularly important for logarithmic functions, which exhibit a decreasing rate of change as the input increases. In simpler terms, they increase rapidly at first but then taper off, increasing more and more slowly as the input grows larger. This characteristic makes them invaluable for modeling scenarios where initial growth is significant but gradually diminishes over time.
For instance, in physics, logarithmic functions appear in the study of entropy, a measure of disorder in a system. The entropy of a system typically increases logarithmically with the number of possible states, reflecting the fact that adding more states has a diminishing impact on the overall disorder. Similarly, in acoustics, the perceived loudness of a sound is proportional to the logarithm of its intensity. This logarithmic relationship explains why we can hear a vast range of sound intensities, from a whisper to a jet engine, without our ears being overwhelmed. The derivative in these contexts helps us understand the rate at which entropy changes or how the perceived loudness changes with respect to sound intensity.
In economics, logarithmic functions are used to model phenomena such as diminishing returns. The law of diminishing returns states that as you add more of one input (e.g., labor) to a fixed amount of other inputs (e.g., capital), the marginal increase in output will eventually decrease. The logarithmic function captures this effect beautifully, showing how the rate of output growth slows down as more input is added. The derivative in this case would help economists analyze the optimal level of input to maximize output.
Computer scientists also leverage logarithmic functions extensively, particularly in the analysis of algorithms. The time complexity of many efficient algorithms, such as binary search, grows logarithmically with the size of the input. This means that the algorithm's runtime increases very slowly as the input size increases, making it highly scalable for large datasets. Understanding the logarithmic nature of the time complexity, often through the use of derivatives in more advanced analyses, is crucial for designing and selecting efficient algorithms.
Furthermore, the derivative of a logarithmic function is inherently linked to the concept of proportional change. Since the derivative of $\ln x$ is $\frac{1}{x}$, a small change in $x$ results in a proportional change in the function's value. This property is fundamental in various applications, including finance, where logarithmic returns are often used to analyze investment performance. The derivative provides a way to quantify and compare these proportional changes across different scenarios.
In conclusion, the exercise of finding the interval of maximum increase for $f(x) = 6 \log_2 x - 3$ is a stepping stone to appreciating the profound impact of logarithmic functions and their derivatives across diverse disciplines. By mastering the techniques for analyzing these functions, we unlock a powerful toolkit for modeling, predicting, and understanding the world around us. The derivative, in this context, is not merely a mathematical tool but a key to unraveling the intricacies of growth, decay, and proportional change in a multitude of real-world phenomena.
In conclusion, the interval over which the function $f(x) = 6 \log_2 x - 3$ increases at the greatest rate is B. [$\frac{1}{8}$, $\frac{1}{2}$]. This conclusion is reached by analyzing the derivative of the function, $f'(x) = \frac{6}{x \ln 2}$, and recognizing its inverse relationship with $x$. The smaller the value of $x$, the larger the value of $f'(x)$, and therefore, the faster the function increases. This problem underscores the importance of understanding derivatives in analyzing the rate of change of functions, particularly logarithmic functions, which have wide-ranging applications in various fields.
