Answer:
C. [tex]\ln z = 5.6157 + i\,\pi[/tex]
Step-by-step explanation:
Let [tex]z = a + i\,b[/tex], [tex]\forall\, a, b\in \mathbb{R}[/tex], the natural logarithm of a this number is:
[tex]\ln z = \ln \|z\|+i\, \theta[/tex] (1)
Where:
[tex]\|z\|[/tex] - Norm of the complex number, no units.
[tex]\theta[/tex] - Direction of the complex number, in radians.
The norm and direction of the complex number are both defined by:
[tex]\|z\| = \sqrt{a^{2}+b^{2}}[/tex] (2)
[tex]\theta = \tan^{-1}\left(\frac{b}{a} \right)[/tex] (3)
If we know that [tex]z = -274.7 +\,i \, 0[/tex], then the natural logarithm of this number is:
[tex]\|z\| = 274.7[/tex], [tex]\theta = \pi\,rad[/tex]
[tex]\ln z = \ln \|z\|+i\, \theta[/tex]
[tex]\ln z = 5.6157 + i\,\pi[/tex]
Hence, correct answer is C.
