Question

The general value of \( \log(1 + i) + \log(1 - i) \) is:

• A. \( \log 2 + 4\pi i \)
• B. \( \log 2 - 4\pi i \)
• C. \( \log 2 + 2\pi i \)
• D. \( \log 3 + 7\pi i \)

Hint:

This question involves the use of logarithmic properties and complex numbers. Always simplify the argument inside the logarithm before applying the logarithmic properties.

Answer 1

The Correct Option is A

We use the logarithmic identity: \[ \log(a) + \log(b) = \log(ab). \] Thus, the expression \( \log(1 + i) + \log(1 - i) \) becomes: \[ \log[(1 + i)(1 - i)] = \log[1^2 - i^2] = \log[1 + 1] = \log 2. \] So, the general value is \( \log 2 \), with the imaginary component arising from the argument of the product of \( (1 + i) \) and \( (1 - i) \), which adds an imaginary factor of \( 4\pi i \).