(2) Find all the values of
a) cosh(log 2),
b) log(log i).

To solve the given problems, we'll use definitions and properties of hyperbolic functions and logarithms, including complex logarithms.

### Part (a): Finding \(\cosh(\log 2)\)

1. **Recall the Definition**: The hyperbolic cosine function \(\cosh x\) is defined as:
   \[
   \cosh x = \frac{e^x + e^{-x}}{2}
   \]
   
2. **Substitute \(x = \log 2\)**:
   \[
   \cosh(\log 2) = \frac{e^{\log 2} + e^{-\log 2}}{2}
   \]
   
3. **Simplify Using the Exponential and Logarithm Properties**:
   \[
   e^{\log 2} = 2 \quad \text{and} \quad e^{-\log 2} = \frac{1}{2}
   \]
   Substituting these into the expression:
   \[
   \cosh(\log 2) = \frac{2 + \frac{1}{2}}{2} = \frac{\frac{4}{2} + \frac{1}{2}}{2} = \frac{\frac{5}{2}}{2} = \frac{5}{4}
   \]

Thus, the value of \(\cosh(\log 2)\) is \(\frac{5}{4}\).

### Part (b): Finding \(\log(\log i)\)

1. **Express \(i\) in Exponential Form**:
   The imaginary unit \(i\) can be expressed as:
   \[
   i = e^{i\frac{\pi}{2}}
   \]
   
2. **Find \(\log i\)**:
   The natural logarithm of \(i\) using the principal branch is:
   \[
   \log i = \log\left(e^{i\frac{\pi}{2}}\right) = i\frac{\pi}{2}
   \]
   
3. **Find \(\log(\log i)\)**:
   Now, we need to find \(\log(i\frac{\pi}{2})\). Using the property of logarithms for a complex number \(z\), we write \(z = re^{i\theta}\), where \(r\) is the magnitude and \(\theta\) is the argument (angle):
   \[
   \log(z) = \log(r) + i\theta
   \]

   - For \(z = i\frac{\pi}{2}\):
     \[
     r = \left|i\frac{\pi}{2}\right| = \frac{\pi}{2}
     \]
     \[
     \theta = \frac{\pi}{2} + 2n\pi \quad \text{(where \( n \) is any integer)}
     \]
     
   - Thus,
     \[
     \log\left(i\frac{\pi}{2}\right) = \log\left(\frac{\pi}{2}\right) + i\left(\frac{\pi}{2} + 2n\pi\right)
     \]
   
4. **Principal Value**: Using \(n = 0\) for the principal value:
   \[
   \log(\log i) = \log\left(\frac{\pi}{2}\right) + i\frac{\pi}{2}
   \]

### Summary of Results:
- \(\cosh(\log 2) = \frac{5}{4}\)
- \(\log(\log i) = \log\left(\frac{\pi}{2}\right) + i\frac{\pi}{2} + 2n\pi i\) (for any integer \( n \))

Do you want more details or have any questions?

### Further Questions:
1. What is the general form of the hyperbolic cosine function?
2. How can we express other trigonometric functions in terms of complex logarithms?
3. What are the properties of complex logarithms that differ from real logarithms?
4. How do hyperbolic functions relate to their trigonometric counterparts?
5. Can the results for \(\log(\log i)\) be generalized for any complex number?

**Tip**: When dealing with logarithms of complex numbers, remember that they have multiple values because of the periodic nature of the complex exponential function.

Explore how to solve cosh(log 2) and log(log i) using hyperbolic functions and complex logarithms. Detailed step-by-step solutions provided with explanations of properties and definitions involved. Suitable for advanced mathematics enthusiasts.

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