Magnitude of complex exponential number, with ln as exponent.

Question

A complex number $ z$ is given: $z = 2\cdot e^{-2\cdot \ln(5)+5\cdot i} $

What is the magnitude of $z$?

Is the answer just:

$ |z| = \sqrt{2^2}=2 $, since the magnitude of the argument is $0$ or am I completely wrong?

Or maybe I can expand $ z$ with cos and sin?

Answer 1

We have that

$$z = 2e^{-2\cdot \ln(5)+5i}=2 \cdot e^{\ln\frac1{25}}\cdot e^{5i}=\frac2{25} e^{5i}$$

which is in the exponential form $z=|z|e^{i\theta}$.