Problem 77 - 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 \(x < 2\).

Step 1: Set the Inner Expression Greater than Zero

In logarithmic function, the argument or the part inside the log function, should always be greater than zero. Hence, set \(2-x > 0\).

Step 2: Solve the Inequality

When solving the inequality \(2 - x > 0\), it results in x < 2.

Step 3: Write Down the Domain of the Function

The solution to the inequality provides all possible 'x' values that 'x' can take, which means it is 'x' is less than 2. So, the domain of the function \(f(x)=\log (2-x)\) is \(x < 2\).

Key Concepts

Domain of a Function

When we talk about the "domain" of a function, we are referring to all the possible input values (usually represented by 'x') that the function can accept to give back a valid output. In the case of logarithmic functions, this concept is especially important because certain rules govern what inputs make sense.

For example, for functions involving logarithms, the argument (the expression inside the logarithm) must be positive. This is because the logarithm of a non-positive number (zero or negative) is not defined in the set of real numbers. Hence, when finding the domain for a logarithmic function like \( f(x)=\log(2-x) \), one starts by setting the expression inside the log greater than zero. This ensures that you only consider values for 'x' that provide a valid argument for the logarithmic function.

In this case, we set \(2-x > 0\). Solving this inequality will then give you the set of all x-values that the function can accept, which forms the domain of the function. Here it results in \( x < 2 \). Thus, the domain is all real numbers less than 2.

Inequalities

Inequalities are mathematical expressions used to compare two values or expressions, indicating one is larger or smaller than the other. They are crucial when finding the domain of functions, such as logarithmic functions, where specific conditions must be met.

To solve an inequality like \(2-x > 0\), you need to isolate 'x' on one side. Start by subtracting 2 from both sides to get \(-x > -2\). Then, multiply both sides by \(-1\) to solve for 'x', remembering to reverse the inequality sign. This results in \(x < 2\).

Having solved the inequality, it tells you the range of values 'x' can take. Here, for the logarithmic function given, only values of 'x' less than 2 are acceptable, ensuring the argument of the log remains greater than zero.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. They form the basis for many mathematical calculations, including determining the domain of functions.

Consider the algebraic expression \(2-x\) from the logarithmic function \(f(x)=\log(2-x)\). This expression dictates the behavior of the logarithmic function. By analyzing this expression, we determine its positivity to find appropriate x-values that the function can accept.

Through manipulating this algebraic expression using inequalities, we solve for the domain of the function. We must consider the conditions that make the argument valid (greater than zero in the case of log functions) and solve accordingly. Thus, understanding algebraic expressions and how to work with them is fundamental in solving for domains and other mathematical problems.