Problem 45 - Precalculus

Question

Find the domain of the function. \(f(x)=\ln (4 x-20)\)

Step-by-Step Solution

Answer

The domain of the function is \( (5, \infty) \).

Step 1: Understand the nature of the logarithmic function

The natural logarithm function, denoted as \(\ln \), is only defined for positive arguments. Therefore, to find the domain of \(f(x) = \ln(4x - 20)\), we need to ensure that the expression inside the logarithm, \((4x - 20)\), is greater than zero.

Step 2: Set up the inequality

To find where \(4x - 20\) is greater than zero, set up the inequality: \[ 4x - 20 > 0 \] This inequality will help us determine the values of \(x\) for which the function is defined.

Step 3: Solve the inequality

Add 20 to both sides of the inequality to isolate the term with \(x\). The modified inequality becomes: \[ 4x > 20 \] Next, divide both sides of the inequality by 4 to solve for \(x\): \[ x > 5 \]

Step 4: State the domain

The solution to the inequality \(x > 5\) tells us the domain of the function. The function \(f(x) = \ln(4x - 20)\) is defined for all real numbers \(x\) that are greater than 5. Therefore, the domain is: \( (5, \infty) \).

Key Concepts

Logarithmic function

A logarithmic function is a fundamental part of algebra and calculus that helps to solve equations involving exponents. Specifically, it's used as the inverse of an exponential function.
For example, if you have the exponential function, such as \(y = b^x\), then the logarithmic function is written as \(x = \log_b(y)\).
Here, \(b\) is the base, which is a positive number other than 1, \(y\) is the input value, and \(x\) is the result. This is useful for many purposes, such as solving equations where the variable is in the exponent or measuring growth and decay processes in natural and social sciences.

• Base 10 Logarithm: If the base is 10, it's called the common logarithm, denoted as \(\log(y)\).
• Natural Logarithm: If the base is the transcendental number \(e\) (approximately 2.718), we use the notation \(\ln(y)\).

Thus, understanding logarithmic functions can be crucial for graphing, transformations, and mathematical modeling tasks.

Inequalities

Inequalities are mathematical expressions used to compare quantities and establish the range of values a variable can hold.
They use symbols such as:

• \(>\) (greater than)
• \(<\) (less than)
• \(\geq\) (greater than or equal to)
• \(\leq\) (less than or equal to)

Solving inequalities involves finding all possible values of the variable that make the inequality true. For example, the inequality \(4x - 20 > 0\) set in the problem shows that we need to determine when the expression inside a logarithm is positive.
To solve it, we perform operations that help isolate the variable, much like solving equations, but we need to be cautious with multiplication or division by negative numbers as they reverse the inequality sign. After isolating \(x\), our task is concluded once we determine the set of values that satisfy the inequality, revealing the domain of the function as \(x > 5\). This means that the inequality defines a specific range of values, crucial for ensuring certain conditions hold in various mathematical contexts.

Natural logarithm

The natural logarithm, denoted as \(\ln\), is a specific type of logarithm with the base \(e\). It is used widely in mathematics due to its unique and useful properties.
One of its key characteristics is that it's only defined for positive values of \(x\). Thus, functions involving \(\ln\) must ensure their arguments are positive, just like in the expression \(f(x) = \ln(4x - 20)\).The base \(e\) is approximately 2.718 and arises naturally in calculations involving continuous growth or compound interest.
Because of its properties, the natural logarithm is particularly useful in calculus for differentiation and integration.

• Properties: The natural logarithm satisfies \(\ln(ab) = \ln(a) + \ln(b)\) and \(\ln(a^b) = b \cdot \ln(a)\).
• Derivatives: For calculus students, \(\ln(x)\) is important because its derivative is \(1/x\), which simplifies many complex calculus problems.

Understanding how to manipulate and apply the natural logarithm is essential for anyone delving into higher mathematics or scientific computations.