Questions: Find the domain of the function. f(x)=6-3 log8[(x/3)-7] The domain of f is (Type your answer in interval notation.)

Solution Steps

To find the domain of the function \( f(x) = 6 - 3 \log_{8}\left[\frac{x}{3} - 7\right] \), we need to determine the values of \( x \) for which the argument of the logarithm is positive. The logarithm function is only defined for positive arguments. Therefore, we solve the inequality \(\frac{x}{3} - 7 > 0\).

To find the domain of the function \( f(x) = 6 - 3 \log_{8}\left[\frac{x}{3} - 7\right] \), we need to determine the values of \( x \) for which the logarithmic expression is defined.

The logarithmic function \(\log_{8}\left[\frac{x}{3} - 7\right]\) is defined only when the argument \(\frac{x}{3} - 7\) is positive. Therefore, we need to solve the inequality:

\[ \frac{x}{3} - 7 > 0 \]

To solve the inequality \(\frac{x}{3} - 7 > 0\), we perform the following steps:

1. Add 7 to both sides of the inequality:

   \[ \frac{x}{3} > 7 \]

2. Multiply both sides by 3 to solve for \( x \):

   \[ x > 21 \]

The solution to the inequality \( x > 21 \) indicates that the domain of the function is all real numbers greater than 21. In interval notation, this is expressed as:

\[ (21, \infty) \]

Final Answer