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

Options

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

Answer

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

### Step-by-Step Solution:

1. **Define the function**: We have \( f(x) = \log(g(x)) \).

2. **Set up the equation**: We need to find \( g(x) \) such that:
\[ g(x) = g(f(x)) \]
Substituting \( f(x) \) into the equation gives:
\[ g(x) = g(\log(g(x))) \]

3. **Assume a form for \( g(x) \)**: Let's assume \( g(x) = e^{h(x)} \) for some function \( h(x) \). This is a common approach since the logarithm and exponential functions are inverses of each other.

4. **Substitute into the equation**: Now substituting \( g(x) \) into our equation:
\[ g(x) = e^{h(x)} \quad \text{and} \quad g(f(x)) = g(\log(g(x))) = g(\log(e^{h(x)})) = g(h(x)) \]
Since \( \log(e^{h(x)}) = h(x) \), we have:
\[ g(h(x)) = e^{h(h(x))} \]

5. **Set the two expressions equal**: Now we have:
\[ e^{h(x)} = e^{h(h(x))} \]
Taking the natural logarithm of both sides gives:
\[ h(x) = h(h(x)) \]

6. **Solve for \( h(x) \)**: The equation \( h(x) = h(h(x)) \) suggests that \( h(x) \) must be a constant function. Let’s denote this constant as \( c \). Therefore, we can write:
\[ h(x) = c \]

7. **Substituting back to find \( g(x) \)**: If \( h(x) = c \), then:
\[ g(x) = e^{h(x)} = e^c \]
Thus, \( g(x) \) is a constant function.

8. **Final conclusion**: Therefore, the value of \( g(x) \) that satisfies \( g(x) = g(f(x)) \) is:
\[ g(x) = e^c \quad \text{for some constant } c. \]

### Answer:
The value of \( g(x) \) is \( e^c \), where \( c \) is a constant.