If `f(x)=logx` and `g(x)=e^(x)`. Find fog and gof , `x gt 0`.

Answer

To solve the problem, we need to find the compositions of the functions \( f \) and \( g \), specifically \( f(g(x)) \) and \( g(f(x)) \).

Given:
- \( f(x) = \log x \)
- \( g(x) = e^x \)

### Step 1: Finding \( f(g(x)) \)

1. **Substitute \( g(x) \) into \( f(x) \)**:
\[ f(g(x)) = f(e^x) \]

2. **Use the definition of \( f(x) \)**:
\[ f(e^x) = \log(e^x) \]

3. **Apply the logarithmic property**:
\[ \log(e^x) = x \cdot \log(e) \]

4. **Since \( \log(e) = 1 \)**:
\[ f(g(x)) = x \cdot 1 = x \]

### Step 2: Finding \( g(f(x)) \)

1. **Substitute \( f(x) \) into \( g(x) \)**:
\[ g(f(x)) = g(\log x) \]

2. **Use the definition of \( g(x) \)**:
\[ g(\log x) = e^{\log x} \]

3. **Apply the property of exponentials and logarithms**:
\[ e^{\log x} = x \]

### Final Results

Thus, we have:
- \( f(g(x)) = x \)
- \( g(f(x)) = x \)

### Summary

Both compositions result in \( x \):
\[ f(g(x)) = x \quad \text{and} \quad g(f(x)) = x \]