Problem 57 - College Algebra

Question

Find the domain of each logarithmic function. $$f(x)=\log (2-x)$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\log (2-x)\) is \(-\infty, 2\).

Step 1: Define the basic logarithm rule

The basic rule for any logarithmic function is that the argument (the part inside the parentheses) must be greater than zero. The reason is that the logarithm function is undefined for non-positive numbers.

Step 2: Apply the rule to the given function

Applying the basic rule of logarithm to the given function \(f(x)=\log (2-x)\), it is required that the argument, \(2-x\), is greater than zero. This leads to the inequality: \(2-x>0\)

Step 3: Solve the inequality

To isolate x, subtract 2 from each side of the equation to obtain the equivalent inequality: \(-x > -2\). Multiplying each side by -1 will change the direction of the inequality, yielding: \(x < 2\). This inequality describes the range of x-values for which \(2 - x > 0\).

Step 4: State the domain

The domain is the solution to inequality. Thus, the domain of the function \(f(x)=\log (2-x)\) is all the x-values that are less than 2. Using interval notation, we could express the domain as \(-\infty, 2\).