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You have two equations to solve for \( x \). Let's go through them one by one.

### (a) \( \ln(\ln(x)) = 1 \)

This equation has already been solved with the correct answer given as:
\[
x = e^e
\]
Explanation:
1. Start by exponentiating both sides of the equation to get rid of the outer natural logarithm:
   \[
   \ln(x) = e^1 = e
   \]
2. Now, exponentiate again to remove the inner natural logarithm:
   \[
   x = e^e
   \]
Thus, the correct solution for part (a) is \( x = e^e \).

### (b) \( e^{ax} = Ce^{bx} \), where \( a \neq b \)

To solve this, follow these steps:

1. Divide both sides by \( e^{bx} \) to simplify:
   \[
   \frac{e^{ax}}{e^{bx}} = C
   \]
2. Using the property of exponents, \( \frac{e^{ax}}{e^{bx}} = e^{(a-b)x} \), this simplifies to:
   \[
   e^{(a-b)x} = C
   \]
3. Now, take the natural logarithm of both sides:
   \[
   (a-b)x = \ln(C)
   \]
4. Finally, solve for \( x \):
   \[
   x = \frac{\ln(C)}{a-b}
   \]

So, the correct answer for part (b) should be:
\[
x = \frac{\ln(C)}{a-b}
\]

This was probably marked wrong because the answer you submitted may have been entered incorrectly or in a different form. Try submitting this expression.

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Would you like further details on any part?  
Here are five related questions to deepen understanding:
1. Why do we exponentiate both sides when solving logarithmic equations?
2. How does the property \( e^{a-b} = e^a / e^b \) help simplify exponent expressions?
3. What is the significance of \( a \neq b \) in equation (b)?
4. Could the equation \( e^{ax} = Ce^{bx} \) have any other forms or interpretations?
5. How do we handle logarithms when \( C \) is negative?

**Tip**: When dealing with exponentials and logarithms, it's essential to remember key properties like \( \ln(e^x) = x \) and the rules for dividing exponents.

Solve each equation for x. (a) ln(ln(x)) = 1, (b) e^ax = Ce^bx, where a ≠ b

Learn how to solve logarithmic and exponential equations step-by-step. In this problem, we solve for x in equations like ln(ln(x)) = 1 and e^ax = Ce^bx, using properties of logarithms and exponentials.

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