Problem 52 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the domain of each logarithmic function. $$f(x)=\ln (x-7)^{2}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \ln (x-7)^2\) is all real numbers, expressed as \((- \infty, \infty)\).

Step 1: Understanding properties of logarithmic functions

Firstly, one must understand that logarithmic functions are defined only for positive arguments. That is, the expression inside the logarithmic function must be greater than zero. So to find the domain of given function, \(f(x) = \ln (x-7)^2\), we need to find for which values of \(x\), the expression \((x-7)^2\) becomes positive.

Step 2: Solving inequality to find the domain

The expression inside the logarithm is \((x-7)^2\). Since any square number is always non-negative, \((x-7)^2\) is always greater than or equal to zero. Thus the function \(f(x)\) is defined for all \(x\).

Step 3: Expressing the Domain

Once we know the function \(f(x)\) is defined for all \(x\), we can express the domain of \(f(x)\) in interval notation. Here, this is the entirety of the real line, or \((- \infty, \infty)\).

Key Concepts

Logarithmic Properties

Unraveling the characteristics of logarithms is crucial in understanding why a function like \(f(x) = \ln(x-7)^2\) has its particular domain.

Logarithmic functions, by their nature, only accept positive arguments to maintain real values. This core principle stems from the definition of a logarithm: if \(b^y = x\), then \(y = \log_b(x)\), where \(x\) must be positive since any real number raised to a power cannot result in a negative number. Other essential properties include:

• \(\log_b(xy) = \log_b(x) + \log_b(y)\), conveying how logs turn multiplication into addition.
• \(\log_b(x/y) = \log_b(x) - \log_b(y)\), showing how division is expressed as subtraction in the logarithmic world.
• \(\log_b(x^y) = y \cdot \log_b(x)\), indicating how exponents transform into coefficients in logs.

Understanding these properties is vital when examining the domain of logarithmic functions, as they reveal the boundaries within which the arguments must fall.

Solving Inequalities

Inequalities present a range of possible solutions, as opposed to the definitive solutions often sought in equations. Solving inequalities is similar to solving equations, but with a particular attention to the direction of the inequality and the operations involved.

When faced with an inequality involving a logarithmic function, such as \(\ln(x-7)^2 > 0\), it's vital to dissect it while preserving the inequality's sense. The rules are straightforward: adding or subtracting the same value on both sides of the inequality does not alter the inequality, same with multiplying or dividing by a positive number; however, things change when you multiply or divide by a negative number, as this flips the inequality direction.

For the function \(f(x) = \ln(x-7)^2\), the inequality step doesn't come into play explicitly, because the square of any real number is non-negative. But, if the inside of the logarithm had been a different expression, solving inequalities would be a central step in determining the function's domain.

Interval Notation

Interval notation is a succinct way to describe sets of numbers along a number line, typically the domain or range of a function. This system uses parentheses and brackets to express intervals:

• Open parentheses, \((\), indicate that an endpoint is not included, akin to a 'less than' (<) or 'greater than' (>) inequality.
• Closed brackets, \([\), show the inclusion of an endpoint, reflecting a 'less than or equal to' (\(\leq\)) or 'greater than or equal to' (\(\geq\)) inequality.

For example, an interval that includes all the numbers greater than 2, but not including 2 itself, is written as \((2, \infty)\). If we were to include 2, it would be written as \([2, \infty)\).

The domain of the function \(f(x) = \ln(x-7)^2\), as derived from its logarithmic properties and the non-negativity of a square, is all real numbers, represented by the interval \((-\infty, \infty)\), a way to say that there are no restrictions on x, as every real number is within the domain of \(f(x)\). This interval includes every possible x-value while adhering to the limitations imposed by the function’s inherent properties.