Logarithm rules for complex numbersAre the logarithm rules true for complex numbers?We know that for...

Sonia Gay

Sonia Gay

Answered

2022-06-24

Logarithm rules for complex numbers
Are the logarithm rules true for complex numbers?
We know that for positive real numbers a, b, c and real number d that:
log b ( a d ) = d log b ( a )
log b ( a ) = log c ( a ) log c ( b )
log b ( x y ) = log b ( x ) + log b ( y )
log b ( x y ) = log b ( x ) log b ( y )
We also know that
log b ( b d ) = d
Does this extend to complex numbers as a, b, c or d?
My instinct is that
log b ( b s + t i ) = s + t i
In other words, I'm pretty confident that the last formula works when d is a complex number

Answer & Explanation

Anika Stevenson

Anika Stevenson

Expert

2022-06-25Added 19 answers

You have to be careful because logs and non-integer powers are multivalued functions. The definition is that a d = exp ( d ln ( a ) ) (for any branch of ln). Now log b ( a d ) is any z such that b z = a d , i.e. exp ( z ln ( b ) ) = exp ( d ln ( a ) ), and that is equivalent to z ln ( b ) d ln ( a ) = 2 π i n for some integer n. So the result is
log b ( a d ) = d ln ( a ) + 2 π i n ln ( b )
Similarly,
log b ( a ) = ln ( a ) + 2 π i m ln ( b )
for some integer m. And thus (assuming you use the same values of ln ( a ) and ln ( b ) in both cases)
log b ( a d ) d log b ( a ) = 2 π i n m d ln ( b )
For example, take a = b = e and d = 2 π i, and use the principal branch of ln
log e ( e 2 π i ) = ln ( 1 ) = 0 but 2 π i log e ( e ) = 2 π i
Another interesting example is a = b = 1, d = 3. Now ( 1 ) 3 = 1, but there is no way to have log 1 ( 1 ) = log 1 ( ( 1 ) 3 ) = 3 log 1 ( 1 ) (this would imply log 1 ( 1 ) = 0, which is certainly false).

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