Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$f(x)=\ln \left(x^{4}+8\right)$$

Step-by-Step Solution

Answer

The domain is all real numbers \((- finity, finity)\).

Step 1: Understand the Logarithm Domain Criteria

The function given is a natural logarithm. Generally, the domain of a logarithmic function, \(f(x) = \ln(g(x))\), is \(g(x) > 0\). This is because the logarithm of non-positive numbers is undefined.

Step 2: Set the Argument Greater Than Zero

To find the domain, set the inside of the logarithm function greater than zero: \(x^4 + 8 > 0\).

Step 3: Solve the Inequality

Since \(x^4 + 8\) is always positive for all \(x\), due to both terms being positive and \(x^4\) never being negative, the inequality holds true for all real numbers. We verify this because any real number \(x\) raised to the fourth power results in a non-negative number.

Step 4: State the Domain

Since \(x^4 + 8 > 0\) for all real numbers, the domain of \(f(x) = \ln(x^4 + 8)\) is all real numbers, expressed in interval notation as \((-finity, finity)\).

Key Concepts

Natural Logarithm

A natural logarithm, often denoted as \(\ln(x)\), is a logarithm whose base is Euler's number \(e\), approximately equal to 2.71828. When you see \(\ln(x)\), it's asking for the power you must raise \(e\) to get \(x\) back again. This logarithm has widespread applications in various fields like calculus, physics, and even complex biological processes.

• The notation \(\ln(x)\) stands for natural logarithm of \(x\).
• The base \(e\) is an irrational and transcendental number.
• The natural logarithm function \(\ln(x)\) is only defined for positive values of \(x\).

Understanding the nature of natural logarithms is important in solving algebraic and calculus problems. It helps in converting multiplication into addition, which often simplifies equations and expressions. Think of \(\ln(x)\) as a tool that turns complexity into simplicity.

Domain of a Function

The domain of a function refers to all the possible input values (usually \(x\)) that will make the function work without any problems like division by zero or taking a logarithm of a non-positive number. For logarithms and specifically the natural logarithm \(\ln(x)\), the domain consists of only positive numbers.In the given example \(f(x) = \ln(x^4 + 8)\), finding the domain involves ensuring the argument \(x^4 + 8 > 0\). Here’s how you'll determine the domain for functions containing a logarithm:

• Find the expression inside the logarithm and set it greater than zero.
• Solve the inequality to determine the values of \(x\) for which the expression holds true.
• For \(\ln(x^4 + 8)\), the expression \(x^4 + 8\) is always greater than zero for any real number \(x\). Thus, the domain is all real numbers, \((-\infty, \infty)\).

This means the function can accept any real number without issue, making it highly adaptable in mathematical modeling and analysis.

Inequality Solving

Inequality solving involves finding all values of a variable that make an inequality true. This concept is fundamental in identifying domains for logarithmic functions, and solving them often requires understanding the behavior of exponents and equations.To solve an inequality like in the function \(f(x) = \ln(x^4 + 8)\):

• Recognize the expression \(x^4 + 8 > 0\) to determine the possible \(x\) values.
• Since any real number raised to a power four, \(x^4\), results in a non-negative number, and adding 8 keeps it non-negative, the inequality \(x^4 + 8 > 0\) holds for all real numbers.
• Therefore, there are no restrictions from the inequality; all \(x\) values satisfy it.

Understanding inequality solving is more than just number crunching; it's about interpreting and understanding mathematical contexts for better decision-making and problem solving.