To find $f(x)$, we need to integrate the given derivative $f'(x) = \frac{1}{x}$ and then apply the condition $f(e) = 2$ to determine the constant of integration.

Step 1: Integrate $f'(x)$

$f(x) = \int \frac{1}{x} \, dx$ The integral of $\frac{1}{x}$ is the natural logarithm: $f(x) = \ln|x| + C$ where $C$ is the constant of integration.

Step 2: Apply the condition $f(e) = 2$

Substitute $x = e$ into the equation: $f(e) = \ln|e| + C = 2$ Since $\ln(e) = 1$, the equation becomes: $1 + C = 2$ Solving for $C$, we get: $C = 1$

Step 3: Final function

Thus, the function $f(x)$ is: $f(x) = \ln|x| + 1$

Would you like more details or have any questions?

Here are five related questions:

1. What are the properties of the natural logarithm function $\ln(x)$?
2. How does the constant of integration affect the solution?
3. What would happen if the condition $f(e) = 2$ were different?
4. Can we determine the behavior of $f(x)$ as $x$ approaches 0 or infinity?
5. How would the solution change if $f'(x)$ were $\frac{2}{x}$ instead of $\frac{1}{x}$?

Tip: Always check initial or boundary conditions after integrating to find the unknown constant!