How do you find the domain of a log function?

To find the domain of a logarithmic function, you need to ensure that the argument (the expression inside the logarithm) is strictly greater than zero. In other words, you solve the inequality: argument > 0.

Here's a breakdown of the process:

1. Identify the argument: Determine the expression that is inside the logarithm. For example, in log(x + 5), the argument is x + 5.

2. Set the argument greater than zero: Write an inequality where the argument is greater than 0. Using the example above, you'd have x + 5 > 0.

3. Solve the inequality: Solve for x. In our example:

   • x + 5 > 0
   • x > -5

4. Express the domain: The solution to the inequality represents the domain of the logarithmic function. You can express this in various ways:

   • Inequality: x > -5
   • Interval Notation: (-5, ∞)
   • Number Line: (As shown in the video excerpt, a number line can visually represent the domain. In this case, an open circle at -5, with the line extending to the right, indicates all values greater than -5 are included.)

Example:

Find the domain of f(x) = log(2x - 6):

1. Argument: 2x - 6
2. Inequality: 2x - 6 > 0
3. Solve:
   • 2x > 6
   • x > 3
4. Domain:
   • Inequality: x > 3
   • Interval Notation: (3, ∞)

In summary, the key is to ensure the expression inside the logarithm is always positive, as logarithms are undefined for zero and negative numbers.