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