Problem 125 - College Algebra

Question

Explain how to find the domain of a logarithmic function.

Step-by-Step Solution

Answer

To find the domain of a logarithmic function \( \log_b(x) \), set the argument of the log function (in this case \( x \)) to be greater than zero. Then, solve the inequality. The domain of the function then is the set of all \( x \) for which \( x > 0 \).

Step 1: Identify the argument of the logarithm

The argument of the logarithm is the expression inside the log function. For example in \( \log_b(x) \), \( x \) is the argument.

Step 2: Set the Argument Greater Than Zero

As previously specified, the argument within the logarithm needs to be strictly greater than zero. So, if the argument is \( x \), give the inequality \( x > 0 \). If the argument is an expression, set the expression to be greater than zero and solve for \( x \)

Step 3: Solve the Inequality

The solution to that inequality will be the domain of the function. Use algebraic techniques such as factoring, completing the square, or using the quadratic formula to solve the inequality, when needed. You may also need to make use of basic knowledge regarding inequalities.

Step 4: Write the Domain

After finding the solution to the inequality, state the domain of the function in interval notation. The domain will be all the \( x \) values for which the argument of the logarithm is strictly greater than zero.