In Exercises 75–80, find the domain of each logarithmic function. f(x) = log₅(x+4)

Verified step by step guidance

1. Recall that the domain of a logarithmic function \( f(x) = \log_b (g(x)) \) consists of all values of \( x \) for which the argument \( g(x) \) is positive, because the logarithm of zero or a negative number is undefined.
2. Identify the argument of the logarithm in the function \( f(x) = \log_5 (x+4) \). Here, the argument is \( x + 4 \).
3. Set the argument greater than zero to find the domain: \( x + 4 > 0 \).
4. Solve the inequality \( x + 4 > 0 \) by subtracting 4 from both sides, which gives \( x > -4 \).
5. Conclude that the domain of \( f(x) = \log_5 (x+4) \) is all real numbers \( x \) such that \( x > -4 \).

Key Concepts

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For logarithmic functions, the domain is restricted because the argument inside the log must be positive. Identifying the domain involves finding all x-values that make the expression inside the logarithm greater than zero.

Properties of Logarithmic Functions

A logarithmic function log_b(x) is defined only for positive arguments x > 0, where b is the base and b > 0, b ≠ 1. This means the expression inside the log must be strictly greater than zero to avoid undefined values. Understanding this property is essential to determine the domain of any logarithmic function.

Inequalities and Solving for Domain

To find the domain of f(x) = log_5(x + 4), solve the inequality x + 4 > 0. This involves basic algebraic manipulation to isolate x and determine the range of values that satisfy the inequality. Solving inequalities is a fundamental skill for defining domains in logarithmic and other functions.