Problem 58 - College Algebra

Question

Find the domain of each logarithmic function. $$f(x)=\log (7-x)$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\log(7-x)\) is all real numbers less than 7, represented as (\(-\infty, 7\)).

Step 1: Understand the properties of logarithm

A critical property of logarithms is that we can only take the logarithm of positive numbers. Therefore, we need to figure out for which values of x, (7-x) is positive.

Step 2: Solve the inequality

To find out where (7-x) is positive, we can set up the inequality \(7-x > 0\). Solving for x, we add x to both sides and subtract 0 from both sides, to get \(x < 7\).

Step 3: Write down the domain

The domain of a function is the set of all possible x-values which will make the function 'work', and will output real y-values. So in this case, the domain is x such that \(x < 7\).