Identify the domain of the following functions: a) $f(x)=\log (-9 x+3)$ Domain: b) $f(x)=\ln (3 x-8)$ Domain: c) $f(x)=\log _{3}(7 x-2)$ Domain: Submit Question

Step 1 ：The domain of a function is the set of all possible input values (often the 'x' variable), which produce a valid output from a particular function. For logarithmic functions, the argument of the logarithm (the expression inside the parentheses) must be greater than zero. This is because logarithms are undefined for zero and negative numbers.

Step 2 ：So, to find the domain of these functions, we need to solve the inequality inside the logarithm > 0.

Step 3 ：For the first function, we need to solve \(-9x + 3 > 0\).

Step 4 ：For the second function, we need to solve \(3x - 8 > 0\).

Step 5 ：For the third function, we need to solve \(7x - 2 > 0\).

Step 6 ：Let's solve these inequalities one by one.

Step 7 ：The solution for the first inequality is \((-\infty, \frac{1}{3})\).

Step 8 ：The solution for the second inequality is \((\frac{8}{3}, \infty)\).

Step 9 ：The solution for the third inequality is \((\frac{2}{7}, \infty)\).

Step 10 ：Final Answer: \[\boxed{\text{a) The domain of } f(x)=\log (-9 x+3) \text{ is } (-\infty, \frac{1}{3})}\]

Step 11 ：\[\boxed{\text{b) The domain of } f(x)=\ln (3 x-8) \text{ is } (\frac{8}{3}, \infty)}\]

Step 12 ：\[\boxed{\text{c) The domain of } f(x)=\log _{3}(7 x-2) \text{ is } (\frac{2}{7}, \infty)}\]