A function `f(x)=log(g(x))`, where `g(x)` is any function of x.
For what value of `g(x)` is the function of `f(x)=g(x)`?

Options

A. a. `g(x)=e`
B. b. `g(x)=e^(x)`
C. c. `g(x)=logx`
D. d. none of these

Answer

To solve the problem, we need to find the value of \( g(x) \) such that the function \( f(x) = g(x) \) holds true, given that \( f(x) = \log(g(x)) \).

### Step-by-Step Solution:

1. **Start with the given function**:
\[ f(x) = \log(g(x)) \]

2. **Set the condition that \( f(x) = g(x) \)**:
\[ g(x) = \log(g(x)) \]

3. **Rearrange the equation**:
\[ g(x) - \log(g(x)) = 0 \]

4. **Let \( y = g(x) \)** for simplicity**:
\[ y - \log(y) = 0 \]

5. **Rearranging gives us**:
\[ y = \log(y) \]

6. **Exponentiate both sides to eliminate the logarithm**:
\[ e^y = y \]

7. **This equation \( e^y = y \) does not have a simple algebraic solution**. However, we can analyze it graphically or numerically. The function \( e^y \) grows exponentially while \( y \) grows linearly. They intersect at only one point.

8. **Finding the intersection**:
- The intersection occurs at \( y = 0 \) (since \( e^0 = 1 \) and \( 0 = \log(1) \)).
- Therefore, \( g(x) = 0 \) is a solution.

9. **Conclusion**:
The only value of \( g(x) \) that satisfies \( f(x) = g(x) \) is:
\[ g(x) = 0 \]