`f(x)=log_(e)x, g(x)=1/(log_(x)e)` . Identical function or not?

Answer

To determine whether the functions \( f(x) = \log_e x \) and \( g(x) = \frac{1}{\log_x e} \) are identical, we need to check three conditions:

1. The domains of both functions must be equal.
2. The ranges of both functions must be equal.
3. For all \( x \) in the common domain, \( f(x) \) must equal \( g(x) \).

Let's analyze these functions step by step.

### Step 1: Determine the domain of \( f(x) \)

The function \( f(x) = \log_e x \) is defined for:
- \( x > 0 \)

Thus, the domain of \( f(x) \) is:
\[ \text{Domain of } f = (0, \infty) \]

### Step 2: Determine the domain of \( g(x) \)

The function \( g(x) = \frac{1}{\log_x e} \) is defined when:
- \( \log_x e \neq 0 \)
- \( x > 0 \)
- \( x \neq 1 \) (since the logarithm is undefined at this point)

Thus, the domain of \( g(x) \) is:
\[ \text{Domain of } g = (0, 1) \cup (1, \infty) \]

### Step 3: Compare the domains

From the above steps, we see:
- Domain of \( f = (0, \infty) \)
- Domain of \( g = (0, 1) \cup (1, \infty) \)

Since the domains are not equal (specifically, \( g(x) \) is undefined at \( x = 1 \)), we conclude that the first condition is not satisfied.

### Step 4: Conclusion

Since the domains of \( f \) and \( g \) are not equal, we can conclude that the functions are not identical.

### Final Answer
The functions \( f(x) \) and \( g(x) \) are **not identical**.

---