Find f(x) given that: f'(x) = 1/x and f(e) =2

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!

Find f(x) given that: f'(x) = 1/x and f(e) = 2

Learn how to solve the problem of finding f(x) given that f'(x) = 1/x and f(e) = 2. This detailed solution uses integration and logarithmic functions to derive the function f(x) = ln|x| + 1. Ideal for students in Grades 11-12.

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