Problem 55 - College Algebra

Question

Find the domain of each logarithmic function. $$f(x)=\log _{5}(x+4)$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\log _{5}(x+4)\) is \(x > -4\).

Step 1: Understand the constraints of a logarithm

Firstly, it is important to understand that the argument of a logarithm has to be greater than 0 (i.e. positive). This is because the logarithm of a negative number or zero is undefined. In our function \(f(x)=\log _{5}(x+4)\), that means that \(x+4\) must be positive.

Step 2: Solve for the domain

Knowing that \(x+4\) must be positive or, in other words, greater than 0, we can now solve for the values of \(x\) that fulfills that requirement. To do this, we can solve the inequality \(x+4 > 0\). Subtracting 4 from each side of the inequality gives us \(x > -4\).

Step 3: Finalize the answer

The solution to the inequality \(x > -4\) tells us that the domain of \(f(x)=\log _{5}(x+4)\) is all real numbers greater than -4. This represents all the x-values which will result in real outputs (y-values) for our function.