Log of negative numbers I know that log of

Dominique Pace 2022-03-21 Answered
Log of negative numbers
I know that log of negative numbers is complex numbers. But I just got over this little proof and wondering what is wrong with this?
log(a)=2×log(a)2=12log(a2)=22log(a)=log(a)
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Answers (2)

Sawyer Anthony
Answered 2022-03-22 Author has 10 answers
Because the property is actually
ln(xa)=aln(|x|)
(for even integer a), and not
ln(xa)=aln(x)
and since no number has a negative absolute value, the second equality is incorrect.
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Alannah Farmer
Answered 2022-03-23 Author has 11 answers
Let's cut the complex plane along the positive real axis and work on the Riemann sheet for which
0argz<2π (1)
For a real-valued with a>0, we have
log(a)=loga+iπ (2)
Now, we have
log(a)=2log(a)2
=12loga2+iπ
=loga+iπ
loga
In fact, if we multiply Equation (2) by a factor of 2, we get
2log(a)=loga+i2πloga
That is, in multiplying by 2, we actually move to the subsequent Riemann surface since Equation (1) restricts the argument to be less than 2π.
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