Log (log i) =

log [ (π)/(2) ] + i [ (π)/(2) ]

log [ (π)/(2) ] - i [ (π)/(2) ]

log ((π)/(2)) - i((π)/(2))

The expression you've provided involves the logarithm of a complex number, specifically log(i). To break it down, we can use the properties of logarithms and the polar form of complex numbers.

Understanding the Components

The complex number \( i \) can be expressed in polar form as:

• Magnitude: \( |i| = 1 \)
• Angle: \( \theta = \frac{\pi}{2} \) (since it lies on the positive imaginary axis)

Logarithm of a Complex Number

The logarithm of a complex number in polar form is given by:

log(z) = log(|z|) + iθ

For \( i \), this becomes:

log(i) = log(1) + i(\frac{\pi}{2})

Since \( log(1) = 0 \), we have:

log(i) = i(\frac{\pi}{2})

Comparing with Your Expression

Your expression seems to involve multiple terms that include \( log(\frac{\pi}{2}) \) and \( i(\frac{\pi}{2}) \). Let's analyze it:

• First term: \( log(\frac{\pi}{2}) \)
• Second term: \( i(\frac{\pi}{2}) log(\frac{\pi}{2}) \)
• Third term: \( -i(\frac{\pi}{2}) log(\frac{\pi}{2}) \)
• Fourth term: \( -i(\frac{\pi}{2}) \)

Combining Terms

When you combine these terms, the imaginary parts involving \( log(\frac{\pi}{2}) \) will cancel out, leading to a simplified expression. The final result will primarily depend on the evaluation of the logarithmic terms.

In summary, the logarithm of \( i \) simplifies to \( i(\frac{\pi}{2}) \), while your expression can be simplified further by combining like terms. If you need further clarification or specific calculations, feel free to ask!