- #1
mepcotterell
- 21
- 0
I know that with base e you use the natural log of complex numbers rule...
ln(x+yi) = ln|x| + arcsin(y/x)i
...because the limit e somehow corresponds to radians about the unit circle
thus, when you take ln(-1) you can do
ln(-1 + 0i) = ln|-1| = arsin(0/-1)i
ln(-1) = 0 + pi*i
ln(-1) = pi*i
My first question is this: What is the relationship between the limit e as the log's base and the unit circle with respect to complex answers?
My second question: How do you find the log of a negative number when the base is not the limit e?
aside... it just dawned upon me while typing this that i might be able to just divide the complex answer by the ln of the desired base? such that...
logb(-1) = ln|-1|/ln(b) + arctan(0/-1)i/ln(b)
logb(-1) = logb|-1| + [1/ln(b)][arctan(0/-1)]i
is that the proper way to do this?
ln(x+yi) = ln|x| + arcsin(y/x)i
...because the limit e somehow corresponds to radians about the unit circle
thus, when you take ln(-1) you can do
ln(-1 + 0i) = ln|-1| = arsin(0/-1)i
ln(-1) = 0 + pi*i
ln(-1) = pi*i
My first question is this: What is the relationship between the limit e as the log's base and the unit circle with respect to complex answers?
My second question: How do you find the log of a negative number when the base is not the limit e?
aside... it just dawned upon me while typing this that i might be able to just divide the complex answer by the ln of the desired base? such that...
logb(-1) = ln|-1|/ln(b) + arctan(0/-1)i/ln(b)
logb(-1) = logb|-1| + [1/ln(b)][arctan(0/-1)]i
is that the proper way to do this?