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

Let's find the values for each part:

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

The hyperbolic cosine function \( \cosh(x) \) is defined as:
\[
\cosh(x) = \frac{e^x + e^{-x}}{2}
\]

Given \( x = \log 2 \), substitute \( x \) into the formula:
\[
\cosh(\log 2) = \frac{e^{\log 2} + e^{-\log 2}}{2}
\]

Now, use the property of logarithms where \( e^{\log x} = x \):
\[
\cosh(\log 2) = \frac{2 + \frac{1}{2}}{2} = \frac{2 + 0.5}{2} = \frac{2.5}{2} = 1.25
\]

Thus, the value of \( \cosh(\log 2) \) is:
\[
\cosh(\log 2) = \frac{5}{4} = 1.25
\]

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

To find \( \log(\log i) \), we first need to evaluate \( \log i \).

1. **Evaluate \( \log i \):**
   
   In the complex plane, the logarithm function is defined as:
   \[
   \log z = \log|z| + i\arg(z)
   \]
   where \( |z| \) is the modulus and \( \arg(z) \) is the argument of the complex number \( z \).

   For \( z = i \), the modulus \( |i| = 1 \), and the argument \( \arg(i) \) is \( \frac{\pi}{2} \) (since \( i \) is on the positive imaginary axis).

   Therefore:
   \[
   \log i = \log 1 + i\frac{\pi}{2} = 0 + i\frac{\pi}{2} = i\frac{\pi}{2}
   \]

2. **Evaluate \( \log(\log i) \):**
   
   Now we need to find \( \log(i\frac{\pi}{2}) \).

   Let \( z = i\frac{\pi}{2} \). Then:
   \[
   \log z = \log\left(\frac{\pi}{2}\right) + i\arg\left(i\frac{\pi}{2}\right)
   \]
   - The modulus \( |z| = \frac{\pi}{2} \).
   - The argument \( \arg(z) = \arg(i) = \frac{\pi}{2} \).

   Thus:
   \[
   \log(i\frac{\pi}{2}) = \log\left(\frac{\pi}{2}\right) + i\frac{\pi}{2}
   \]

So, the value of \( \log(\log i) \) is:
\[
\log\left(i\frac{\pi}{2}\right) = \log\left(\frac{\pi}{2}\right) + i\frac{\pi}{2}
\]

### Summary of Results:

- **(a) \( \cosh(\log 2) = 1.25 \)**
- **(b) \( \log(\log i) = \log\left(\frac{\pi}{2}\right) + i\frac{\pi}{2} \)**

Would you like more details on any of these steps or have any further questions? Here are a few related questions:

1. How do the hyperbolic functions compare to their trigonometric counterparts?
2. What is the geometric interpretation of the complex logarithm function?
3. How does the argument of a complex number affect its logarithm?
4. Can \( \cosh(x) \) ever be negative? Why or why not?
5. What are the branches of the logarithm function in the complex plane?

**Tip:** When working with complex logarithms, always remember that the function is multivalued due to the periodicity of the argument. Understanding this will help avoid potential pitfalls.

Learn how to calculate the values of cosh(log 2) and log(log i) step-by-step with detailed explanations. This problem involves hyperbolic functions and complex logarithms, suitable for advanced undergraduate students.

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