How to calculate the common logarithm of a complex number?

To calculate the common logarithm of a complex number, we need to convert it to polar form.

First, we can write the complex number in the form a + bi, where a and b are real numbers and i is the imaginary unit. Then, we can find the modulus (r) and argument (θ) of the complex number using the following formulas:

r = √(a^2 + b^2)
θ = tan^-1(b/a)

Next, we can write the complex number in polar form as r(cosθ + isinθ). Finally, we can use the following formula to find the common logarithm of the complex number:

log(z) = log(r) + iθ

For example, let's find the common logarithm of the complex number 2 + 3i:

r = √(2^2 + 3^2) = √13
θ = tan^-1(3/2) = 56.31°

Therefore, the complex number in polar form is √13(cos56.31° + isin56.31°). Using the formula above, we can find the common logarithm:

log(2 + 3i) = log(√13) + i(56.31°/log(10))
= 0.1139 + i0.8686

Therefore, the common logarithm of 2 + 3i is approximately 0.1139 + i0.8686.