Final answer:
To calculate Arg z and log z, we can use formulas and the given complex numbers. For -√3+i, Arg z = -π/6 and log z = log(2) + i * (-π/6). For (√3+1)²⁰⁰¹, Arg z = π/6 and log z = log(2) + i * (π/6).
Step-by-step explanation:
To calculate Arg z, we need to find the angle that the complex number makes with the positive x-axis in the complex plane. This can be done using the formula: Arg z = arctan(Cy/Cx). For the complex number -√3+i, we have Cx = -√3 and Cy = 1. Substituting these values into the formula, Arg z = arctan(1/(-√3)) = arctan(-1/√3) = -π/6.
To calculate log z, we need to find the logarithm of the complex number. Using the formula: log z = log(|z|) + i * Arg z, where |z| is the modulus of z. For the complex number -√3+i, |z| = √((-√3)² + 1²) = √(3+1) = √4 = 2. Thus, log z = log(2) + i * (-π/6).
For the complex number (√3+1)²⁰⁰¹, we can calculate the Arg z using the same formula. Cx = √3 and Cy = 1. Substituting these values, Arg z = arctan(1/√3) = π/6. To calculate log z, we first need to find the modulus |z|, which is √((√3)² + 1²) = √(3+1) = √4 = 2. Thus, log z = log(2) + i * (π/6).