Problem 125 - College Algebra Essentials

Question

Explain how to find the domain of a logarithmic function.

Step-by-Step Solution

Answer

The domain of a logarithmic function is \((0, +\infty)\), meaning all real numbers greater than zero.

Step 1: Understand the logarithm function

A logarithmic function has the form \(f(x) = \log_b (x)\), where \(b > 0\), \(b \neq 1\), and \(x > 0\). The base \(b\) is usually seen as 10 (common logarithm) or \(e\) (natural logarithm). The logarithm function is the inverse of the exponential function.

Step 2: Identify the domain of the function

The domain of a function is the set of all possible input values (typically the 'x' variable). For a logarithmic function, it refers to the set of all x-values that will output real numbers.

Step 3: Apply the rule

From the natural logarithm function, as a specific case, we know that \(ln(0)\) is undefined. Meanwhile, \(ln(x)\) gives real numbers for any \(x > 0\). Therefore, the domain of a logarithmic function is \((0, +\infty)\), meaning it’s defined for all real numbers greater than zero.