Transcript text: Find the domain of $y=\log (6-2 x)$.
Solution
Solution Steps
Step 1: Identify the Argument
The argument of the logarithm is given by \(ax + b\).
Step 2: Determine the Constraint
For the logarithm to be defined, its argument must be greater than zero: \(ax + b > 0\).
Step 3: Solve the Inequality
Given that \(a = -2\) and \(b = 6\), we solve the inequality \(ax + b > 0\).
Since \(a < 0\), the solution to the inequality is \(x < -\frac{6}{-2}\).
This simplifies to ax + b > 0 \Rightarrow x < -\frac{6}{-2}.
Step 4: Express the Domain
The domain of \(y = \log(ax + b)\) is given by the solution to the inequality, which is x < 3 in interval notation.
Final Answer:
The domain of the function \(y = \log(ax + b)\) is x < 3, where \(a = -2\) and \(b = 6\).