f(x) = `log_(x) 10` is defined in the domain

Options

A. `x gt 10`
B. `x gt 10`. Except x = 10
C. `x ge 10`
D. `x gt 0`. Except x = 1

Answer

To determine the domain of the function \( f(x) = \log_{x} 10 \), we need to consider the properties of logarithms. Here’s the step-by-step solution:

### Step 1: Identify the conditions for the logarithm
The logarithm function \( \log_{b}(a) \) is defined under certain conditions:
1. The base \( b \) must be a positive number.
2. The base \( b \) cannot be equal to 1.
3. The argument \( a \) must be a positive number.

### Step 2: Apply the conditions to our function
In our case, the function is \( f(x) = \log_{x} 10 \). Here:
- The base \( x \) must be greater than 0 (i.e., \( x > 0 \)).
- The base \( x \) cannot be equal to 1 (i.e., \( x \neq 1 \)).
- The argument 10 is already a positive number, so it satisfies the condition.

### Step 3: Combine the conditions
From the conditions above, we can summarize the domain of \( f(x) \):
- \( x > 0 \) (to ensure the base is positive)
- \( x \neq 1 \) (to ensure the base is not equal to 1)

Thus, the domain can be expressed as:
\[ x \in (0, 1) \cup (1, \infty) \]

### Conclusion
The correct option that describes the domain of \( f(x) = \log_{x} 10 \) is:
- \( x > 0 \) except \( x = 1 \)