Find fog and gof if:
`f(x)=e^(x),g(x)=lnx`

Explanation

To find \( f \circ g \) and \( g \circ f \) for the functions \( f(x) = e^x \) and \( g(x) = \ln x \), we will follow these steps:

### Step 1: Find \( f \circ g \)

1. **Definition of Composition**: The composition \( f \circ g \) means \( f(g(x)) \).
2. **Substituting \( g(x) \)**: We know \( g(x) = \ln x \). So, we need to find \( f(g(x)) = f(\ln x) \).
3. **Using the Function \( f \)**: Now substitute \( \ln x \) into \( f(x) \):
\[ f(\ln x) = e^{\ln x} \]
4. **Simplifying**: Using the property of logarithms that states \( e^{\ln a} = a \) (for \( a > 0 \)):
\[ e^{\ln x} = x \]
5. **Final Result for \( f \circ g \)**:
\[ f \circ g = x \quad \text{for } x > 0 \]

### Step 2: Find \( g \circ f \)

1. **Definition of Composition**: The composition \( g \circ f \) means \( g(f(x)) \).
2. **Substituting \( f(x) \)**: We know \( f(x) = e^x \). So, we need to find \( g(f(x)) = g(e^x) \).
3. **Using the Function \( g \)**: Now substitute \( e^x \) into \( g(x) \):
\[ g(e^x) = \ln(e^x) \]
4. **Simplifying**: Using the property of logarithms that states \( \ln(a^b) = b \ln a \):
\[ \ln(e^x) = x \cdot \ln e \]
Since \( \ln e = 1 \):
\[ \ln(e^x) = x \]
5. **Final Result for \( g \circ f \)**:
\[ g \circ f = x \]

### Summary of Results:
- \( f \circ g = x \) for \( x > 0 \)
- \( g \circ f = x \)