How do you evaluate log4(164)?

1 Answer
Aug 9, 2016

log4(164)=3+iπ±2kπloge4

for k=0,1,2,

Explanation:

Let us investigate the complex solutions. We know by the logarithm definition

4z=164=43

Supposing now z=x+iy we have

4x4iy=43 so we have

4x=43x=3 and
4iy=1

We know

4=eloge4 so

4iy=eiyloge4=cos(yloge4)+isin(yloge4)=1. This condition is attained for

yloge4=π±2kπ with k=0,1,2,3,4,

so

y=π±2kπloge4

Finally

log4(164)=3+iπ±2kπloge4