The domain of the function f defined by `f(x)=log_(e)(5-6x)` is

Options

A. `(-oo,(5)/(6))`
B. `((5)/(6),oo)`
C. `(-oo,(5)/(6)]`
D. `[(5)/(6),oo)`

Answer

To find the domain of the function \( f(x) = \log_e(5 - 6x) \), we need to determine the values of \( x \) for which the expression inside the logarithm is positive, since the logarithm of zero or a negative number is undefined.

### Step-by-Step Solution:

1. **Set the argument of the logarithm greater than zero**:
\[ 5 - 6x > 0 \]

2. **Rearrange the inequality**:
\[ 5 > 6x \]
or equivalently,
\[ 6x < 5 \]

3. **Divide both sides by 6**:
\[ x < \frac{5}{6} \]

4. **Conclusion about the domain**:
Since there are no other restrictions on \( x \), the domain of the function \( f(x) \) is:
\[ x < \frac{5}{6} \]
In interval notation, this can be expressed as:
\[ (-\infty, \frac{5}{6}) \]

### Final Answer:
The domain of the function \( f(x) = \log_e(5 - 6x) \) is \( (-\infty, \frac{5}{6}) \).