Problem 77 - Algebra and Trigonometry

Question

In Exercises 75–80, 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 argument greater than zero

Based on the property of logarithmic function, we know the argument must be greater than 0. Therefore, we set up an inequality \(2 - x > 0\).

Step 2: Solve the inequality

To find the domain of the function, solve this inequality for x. We can do so by subtracting 2 from both sides of the inequality to isolate the x term, which gives us \(-x > -2\). Then, multiply both sides by -1 (which reverses the direction of the inequality) to be \(x < 2\).

Step 3: State the domain

The solution to the inequality \(x < 2\) is our domain. Therefore, the domain of the function \(f(x) = \log(2 - x)\) is all real numbers less than 2.

Key Concepts

Logarithmic Function Properties

Understanding the properties of logarithmic functions is crucial for solving problems related to them. A logarithmic function is the inverse of an exponential function. It is defined only for positive real numbers as it represents the power to which a fixed base must be raised to produce a given number. This is why when setting up an expression such as \( f(x) = \log(2 - x) \), the argument \( 2 - x \) must be greater than zero.

Several key properties define how logarithms behave:

• The base of a logarithm must be a positive real number not equal to 1.
• The logarithm of 1 to any base is always 0, because any number raised to zero power is 1.
• Logarithms can be used to solve exponential equations because of their property of transforming multiplication into addition, and powers into products.

When we solve for the domain, we're using these properties to determine the set of all possible inputs that will make the function valid and define its scope.

Solving Inequalities

Inequalities are comparisons between two expressions that may not be equal but instead have a greater than or less than relationship. Solving inequalities is similar to solving equations, but one must remember that multiplying or dividing both sides by a negative number reverses the inequality sign.

Here's a step-by-step method to solve an inequality such as \( 2 - x > 0 \):

• Isolate the variable on one side of the inequality.
• If multiplying or dividing by a negative number, flip the direction of the inequality.
• The solution set includes all values that satisfy the inequality, graphically representable on a number line.

In the case of the logarithmic inequality from the exercise, isolating \( x \) involves reversing the inequality when we multiply by -1. This process results in the domain \( x < 2 \). Being meticulous in adhering to these steps ensures an accurate solution.

Function Domain

The domain of a function is the complete set of possible values of the independent variable which will output real numbers from a given function. For logarithmic functions, the domain involves finding the values that the input argument can take without violating the base conditions of logarithms, namely that the argument must be positive.

The process for determining the domain includes:

• Setting the inside of the logarithmic function greater than zero.
• Solving the resulting inequality.
• Expressing the solution in terms of all acceptable input values.

In our exercise, the domain of \( f(x) = \log(2 - x) \) is calculated by ensuring the expression inside the logarithm is positive. Hence, we solve the inequality \( 2 - x > 0 \) leading us to the domain: the set of all real numbers less than 2.