What is the logarithm of -1
Everyone knows that logarithm of 1 is zero.
But 1=(-1)^2
So ln( (-1)^2 ) must equal zero
but ln a^b = b*ln(a)
So 2*ln(-1) must equal zero
or ln(-1)=0
Quote: pacomartin...
but ln a^b = b*ln(a)
...
Can you prove that line is true for a and b as negative numbers? I think if your conclusion were correct, then you are working toward ln(x)=ln(-x) in something like this:
ln(-x)=ln(x*(-1))=ln(x)+ln(-1)=ln(x)+0=ln(x)
If that were true and y=ln(x), then ey=x and ey=-x
I think I'd instead go for:
ln(-1)=iπ
The definition of "natural logarithm of x" is the number y such that ey = x.
eiπ = -1; therefore, ln -1 = iπ.
Quote: ThatDonGuyThe definition of "natural logarithm of x" is the number y such that ey = x.
eiπ = -1; therefore, ln -1 = iπ.
Then also ln(-1) = -i*pi, since exp(-i*pi) = -1. Or ln(-1) = 3i*pi. The usual definition of the natural logarithm does not help.
In fact a common definition could be ln(x) = ln(|x|) + i arg(x) for complex x, but this has a discontinuity for negative x.
Quote: Doc...
I think if your conclusion were correct, then you are working toward ln(x)=ln(-x) in something like this:
....
I think I'd instead go for:ln(-1)=iπ
In reality Leonard Euler did the exact same thing in 1727 and concluded that his argument led to ln(x)=ln(-x) . He eventually worked out the correct answer which he published in 1740 in his book Introductio in analysin infinitorum.
You are too hard on yourself Doc. If you find yourself repeating the exact same process as one of the world's great thinkers than you should pat yourself on the back.
Hence, writing ln 1 = 0 is valid only in the real domain. In the complex plane, e^x is not bijective, ln x has several "branches".
Quote: kubikulannIn the complex plane, e^x is not bijective, ln x has several "branches".
It does not seem that Bernoulli figured out that the function was bijective. Euler published the final formula when Bernoulli was very old.
Quote: Ed SandiferThe English mathematician Roger Cotes (1862-1716) was studying problems in the arc length of spirals. In about 1712, in the course of his investigations, he seems to be the first one to discover a formula equivalent to the Euler formula:
ln (cos x +i sin x) = ix
This is easily transformed into the Euler formula by exponentiating both sides, but apparently Cotes never did this. Moreover, Cotes died rather suddenly in 1716 without publishing much of his work on this subject.
It seems like a trivial matter to go from the Cotes formula to the Euler formula. It gives you an appreciation for the insight we take for granted today in inverse functions.
