ln ( 10.75 + 6.98i )

ln ( 10.75 + 6.98i )

$$\small{\text{ $z=10.75+6.98\ i \quad \ln{(z)} = ?$ }}\\ \small{ Solution: \boxed{ln{(z)} = \ln{(|z|)} + i*(arg(z)+2\pi k ) } } \\\\ \small{\text{ I. $ |z| = \sqrt{10.75^2+6.98^2} = 12.8172891050 $ }}\\ \small{\text{ II. $ \phi\ensurement{^{\circ}} = \tan^{-1}{(\frac{6.98}{10.75} )} = 32.9957575073\ensurement{^{\circ}} $ }}\\ \small{\text{ III. $ arg(z) = \phi \ensurement{^{\circ}} * \frac{ \pi }{ 180\ensurement{^{\circ}} }+2\pi k = 0.57588460769 + 2\pi k$ }}\\\\ \small{\text{ $ \ln{(z)} = \ln{(12.8172891050 )}+i*(0.57588460769+2\pi k) $ }}\\ \small{\text{ $ \boxed{ \ln{(10.75+6.08\ i)} = 2.55079497086+(0.57588460769+2\pi k) \ i } \quad k=0,1,2\dots $ }}$$

Thanks Heureka, that looks impressive  

Here's an alternative approach (I've just considered the principal solution):

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