Problem 56 - College Algebra

Question

Find the domain of each logarithmic function. $$f(x)=\log _{5}(x+6)$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\log _{5}(x+6)\) is all real numbers \(x\) such that \(x > -6\).

Step 1: Identify the Argument of the Logarithm

The argument of the logarithm is \(x+6\) here. This is the expression that needs to be greater than 0 for the logarithm to be defined.

Step 2: Set up the Inequality

The argument of the logarithmic function must be greater than zero. So, we can set up the inequality as \(x + 6 > 0\).

Step 3: Solve the Inequality

Subtract 6 from both sides of the inequality to isolate \(x\), resulting in the inequality: \(x > -6\).