if f(x)=ln x and g(x)= e^(x+1) find f(g(2))- g (f(e))

Let's solve the expression \( f(g(2)) - g(f(e)) \) given the functions \( f(x) = \ln(x) \) and \( g(x) = e^{x+1} \).

### Step 1: Calculate \( g(2) \)
Given \( g(x) = e^{x+1} \), we substitute \( x = 2 \):
\[
g(2) = e^{2+1} = e^3
\]

### Step 2: Calculate \( f(g(2)) \)
Now, substitute \( g(2) = e^3 \) into the function \( f(x) = \ln(x) \):
\[
f(g(2)) = f(e^3) = \ln(e^3)
\]
Since \( \ln(e^3) = 3 \ln(e) \) and \( \ln(e) = 1 \), we get:
\[
f(g(2)) = 3
\]

### Step 3: Calculate \( f(e) \)
Substitute \( x = e \) into \( f(x) = \ln(x) \):
\[
f(e) = \ln(e) = 1
\]

### Step 4: Calculate \( g(f(e)) \)
Now, substitute \( f(e) = 1 \) into the function \( g(x) = e^{x+1} \):
\[
g(f(e)) = g(1) = e^{1+1} = e^2
\]

### Step 5: Calculate \( f(g(2)) - g(f(e)) \)
Subtract the results from Step 2 and Step 4:
\[
f(g(2)) - g(f(e)) = 3 - e^2
\]

So, the final answer is:
\[
\boxed{3 - e^2}
\]

Would you like further details or have any questions?

Here are some related questions you might find helpful:

1. What is the derivative of \( f(x) = \ln(x) \) and \( g(x) = e^{x+1} \)?
2. How would the solution change if \( g(x) = e^{2x+1} \) instead?
3. What happens if you substitute different values for \( x \) in the expression \( f(g(x)) - g(f(x)) \)?
4. Can you find the inverse functions for \( f(x) \) and \( g(x) \)?
5. How would you solve \( f(g(x)) = g(f(x)) \) for a general \( x \)?

**Tip:** When working with logarithmic and exponential functions, remember that they are inverses of each other, which often simplifies the calculations.

Learn how to solve the mathematical expression f(g(2)) - g(f(e)) using the functions f(x) = ln(x) and g(x) = e^(x+1). This problem involves calculating values using logarithmic and exponential functions, suitable for advanced high school students.

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