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Chapter 7: Problem 6

For each function, find the domain. $$ f(x, y)=\frac{x}{\ln y} $$

Short Answer

Expert verified
The domain is \( \{ (x, y) \mid y \in (0,1) \cup (1,\infty) \} \).

Step by step solution

01

Identify Restrictions from the Logarithm

The natural logarithm, \(\ln y\), is only defined for positive values of \(y\). Therefore, for the function \(f(x, y) = \frac{x}{\ln y}\) to be defined, \(y > 0\). Additionally, \(\ln y = 0\) would make the function undefined due to division by zero, which happens when \(y = 1\). Thus, \(y\) must be positive and not equal to 1.
02

Define the Domain Constraints

From the restrictions determined: \(y > 0\) and \(y eq 1\). \(x\) can take any real number value, as there are no restrictions imposed on \(x\) by the function.
03

Write the Domain

The domain considers both constraints together. Therefore, the domain of \(f(x, y)\) is all real numbers \(x\) paired with \(y\) such that \(y \in (0, 1) \cup (1, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Natural Logarithm
The natural logarithm, denoted as \(\ln(y)\), is a powerful mathematical function often used in calculus and algebra. Unlike regular logarithms with arbitrary bases, the natural logarithm uses \(e\), approximately 2.718, as its base. It is essential because it helps convert multiplicative relationships into additive ones, simplifying complex calculations.
In our specific exercise involving the function \(f(x, y) = \frac{x}{\ln y}\), the natural logarithm introduces unique considerations:
  • \(\ln(y)\) is only defined when \(y > 0\) because logarithms of non-positive numbers are not real.
  • The expression becomes \(\frac{x}{\ln y}\), making the function undefined when \(\ln(y) = 0\), which occurs at \(y = 1\).
The magic of \(\ln(y)\) lies in its fundamental properties, allowing us to explore areas such as calculus, growth rates, and information theory.
Recognizing Restrictions in Functions
When dealing with functions, recognizing any imposed restrictions is crucial for finding the domain and ensuring the function remains valid. Restrictions occur when there are limits on the input values, making sure there are no undefined operations or problematic calculations.
For the function \(f(x, y) = \frac{x}{\ln y}\), we immediately identify:
  • \(y > 0\): Because \(\ln(y)\) requires a positive input, ensuring the calculation is real.
  • \(y eq 1\): As \(\ln(1) = 0\), division by zero would render the function undefined. Thus, \(y\) must never equal 1.
These constraints influence the permissible values of \(y\), shaping the domain for the function's use.
Mastering Calculus Problem Solving
Calculus problem solving involves using mathematical reasoning to navigate through function properties and restrictions. In the context of our function \(f(x, y) = \frac{x}{\ln y}\), properly understanding and applying constraints is key to finding an accurate domain.
Here's how we apply these principles:
  • Identify function properties, such as division and logarithms, to anticipate restrictions.
  • Analyze each component (i.e., \(\ln y\)) to ensure they meet the necessary conditions.
  • Combine all conditions transparently to create a comprehensive domain, such as \(y \in (0, 1) \cup (1, \infty)\).
By breaking down the problem, defining constraints, and methodically progressing step-by-step, solving calculus problems becomes an approachable task, bringing clarity to complex functions.

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