Selecting The Correct Increasing Logarithmic Function

In the realm of mathematics, particularly when dealing with logarithmic functions, it's crucial to understand how the function's parameters affect its behavior. This article delves into the process of identifying an increasing logarithmic function with a specific domain. We'll analyze the given options, focusing on the transformations applied to the basic logarithmic function and their impact on the domain and increasing/decreasing nature of the function. Understanding these nuances is essential for anyone studying calculus, pre-calculus, or related fields. This article will not only provide the solution but also explain the underlying principles, helping you to tackle similar problems with confidence.

Understanding Logarithmic Functions

Before we dive into the specific problem, let's establish a solid foundation by understanding the characteristics of logarithmic functions. The basic logarithmic function, typically written as $f(x) = \log_b(x)$, where $b$ is the base (and usually $b > 1$ for introductory calculus), has several key properties. The domain of this function is $(0, \infty)$, meaning it's only defined for positive values of $x$. It's an increasing function, which means that as $x$ increases, $f(x)$ also increases. The graph of the basic logarithmic function has a vertical asymptote at $x = 0$, and it passes through the point $(1, 0)$. Transformations applied to this basic function, such as horizontal and vertical shifts, reflections, and stretches/compressions, will alter these characteristics. When determining whether a logarithmic function is increasing or decreasing, one must pay close attention to the sign of the coefficient of the logarithmic term and any reflections across the x-axis. For instance, if the logarithmic term is multiplied by a negative number, the function will be decreasing instead of increasing. Furthermore, horizontal shifts affect the domain of the function, while vertical shifts simply move the graph up or down without changing the domain's width or the increasing/decreasing nature.

Analyzing the Given Options

Now, let's consider the options provided in the question. We need to find an increasing logarithmic function with a domain of $(1, \infty)$. This means the function should be defined for all $x$ values greater than 1. Let's examine each option:

A. $f(x) = -\log(x-2) + 1$

In this option, we see a logarithmic function with a horizontal shift and a reflection. The (x-2) inside the logarithm indicates a horizontal shift to the right by 2 units. This means the domain of this function will be $(2, \infty)$, since the argument of the logarithm (x-2) must be greater than 0. The negative sign in front of the logarithm indicates a reflection across the x-axis, which means this function is decreasing, not increasing. Therefore, option A is not the correct answer.

B. $f(x) = -\log(x-1) + 2$

Similar to option A, this function also has a horizontal shift and a reflection. The (x-1) inside the logarithm indicates a horizontal shift to the right by 1 unit, making the domain $(1, \infty)$. However, the negative sign in front of the logarithm again indicates a reflection across the x-axis, making this function decreasing. Therefore, option B is also not the correct answer.

C. $f(x) = \log(x-2) + 1$

This option presents a logarithmic function with a horizontal shift but no reflection. The (x-2) inside the logarithm indicates a horizontal shift to the right by 2 units, meaning the domain is $(2, \infty)$. Since there is no negative sign in front of the logarithm, this function is increasing. However, the domain does not match the required domain of $(1, \infty)$, so option C is not the correct answer.

D. $f(x) = \log(x-1) + 2$

In this option, we have a logarithmic function with a horizontal shift but no reflection. The (x-1) inside the logarithm indicates a horizontal shift to the right by 1 unit, giving us a domain of $(1, \infty)$. Since there is no negative sign in front of the logarithm, this function is increasing. This option satisfies both the increasing condition and the domain requirement.

The Correct Answer and Detailed Explanation

Based on our analysis, the correct answer is D. $f(x) = \log(x-1) + 2$. Let's delve deeper into why this is the correct answer. This function is a logarithmic function with a base of 10 (since no base is explicitly written, it's assumed to be 10), shifted 1 unit to the right and 2 units upwards. The horizontal shift is determined by the (x-1) term inside the logarithm, which shifts the graph 1 unit to the right. This shift also changes the vertical asymptote from $x = 0$ (in the basic logarithmic function) to $x = 1$. Therefore, the domain of the function becomes $(1, \infty)$, which matches the specified requirement. The vertical shift of 2 units upwards simply moves the entire graph up by 2 units, but it does not affect the domain or whether the function is increasing or decreasing. The most crucial aspect is the absence of a negative sign in front of the logarithm. This ensures that the function retains its increasing nature. As $x$ increases within the domain $(1, \infty)$, the value of (x-1) also increases, and consequently, the value of $\log(x-1)$ increases. Adding 2 to this increasing value does not change the increasing nature of the function.

To further illustrate this, consider two values of $x$ within the domain, say $x = 2$ and $x = 11$. When $x = 2$, $f(2) = \log(2-1) + 2 = \log(1) + 2 = 0 + 2 = 2$. When $x = 11$, $f(11) = \log(11-1) + 2 = \log(10) + 2 = 1 + 2 = 3$. Since $f(11) > f(2)$ when $11 > 2$, this demonstrates that the function is indeed increasing. This detailed explanation solidifies the understanding of why option D is the correct answer and how the transformations applied to the basic logarithmic function affect its properties.

Key Takeaways and General Strategies

In summary, when analyzing logarithmic functions to determine their increasing/decreasing nature and domain, consider the following key takeaways:

• Domain: The argument of the logarithm (the expression inside the logarithm) must be greater than 0. This is a fundamental rule for determining the domain of logarithmic functions. Horizontal shifts directly affect the domain. For example, if you have $\log(x-a)$, the domain will be $(a, \infty)$.
• Increasing/Decreasing: The sign of the coefficient of the logarithmic term determines whether the function is increasing or decreasing. If the coefficient is positive, the function is increasing; if it's negative, the function is decreasing (due to a reflection across the x-axis).
• Transformations: Understand how transformations, such as horizontal and vertical shifts, reflections, and stretches/compressions, affect the graph and properties of the logarithmic function.
• Vertical Shifts: Vertical shifts do not affect the domain or the increasing/decreasing nature of the function. They simply move the graph up or down.
• Reflection: A negative sign in front of the logarithmic term signifies a reflection across the x-axis, which inverts the increasing/decreasing nature of the function.

When tackling similar problems, it's helpful to first identify the transformations applied to the basic logarithmic function. Then, determine the domain by setting the argument of the logarithm greater than 0. Finally, check the sign of the coefficient of the logarithmic term to determine if the function is increasing or decreasing. If you can systematically analyze these factors, you'll be well-equipped to solve a wide range of problems involving logarithmic functions. This strategic approach not only helps in finding the correct answer but also enhances your understanding of the underlying mathematical principles, making you a more proficient problem-solver in mathematics and related fields. Practice applying these strategies to various problems to solidify your understanding and build confidence in your abilities.