Find the Natural Logarithm of -1-i: A Step-by-Step Guide

To find the natural logarithm of -1-i, we can write -1-i in polar form.

Let's represent -1-i as z = r * e^(iθ), where r is the magnitude of z and θ is the argument of z.

To find r, we use the formula r = √((-1)^2 + (-1)^2) = √(1 + 1) = √2.

To find θ, we use the formula θ = arctan((-1)/(-1)) = arctan(1) = π/4.

Therefore, z = √2 * e^(iπ/4).

Now, we can use the property of the natural logarithm that ln(a * b) = ln(a) + ln(b).

ln(-1-i) = ln(√2 * e^(iπ/4))

Using the property ln(a * b) = ln(a) + ln(b), we can rewrite this as:

ln(√2) + ln(e^(iπ/4))

Now, ln(√2) is a real number and can be calculated using a calculator or logarithm tables.

Finally, we have:

ln(-1-i) = ln(√2) + ln(e^(iπ/4))

Please note that ln(a) represents the natural logarithm of a.