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Who was the first person to notice logarithms of negatives numbers and why? When did they first arise naturally? I thought I saw somewhere that it had to do with integration but I can't find it anymore and it's really bothering me. What was this integral (if indeed that's when mathematicians first stumbled upon logarithms of negatives)?


Also, this is somewhat of a follow-up question to: When and WHY did mathematicians start turning their attention to imaginary exponents?

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As far as I known, logarithms of negative numbers first appeared (in the modern sense) in the 1751 paper of Euler De la controverse entre Mrs. Leibniz et Bernoulli sur les logarithmes des nombres negatifs et imaginaires, you can find here the original and the English translation. As you will see, Euler discuss the different positions of Bernoulli and Leibniz about this kind of quantities, so of course they considered logarithms of negative numbers before Euler. Their correspondence from 1694 to 1716 was published in 1745 as the two-volume collection Virorum celeberr. Got. Gul. Leibnitii et Johan. Bernoullii Commercium philosophicum et mathematicum.

(Euler write $l-a$ for $\log(-a)$)

POSITION OF M. BERNOULLI

M. Bernoulli holds that the logarithms of negative numbers are the same as those of positive numbers, in other words that the logarithm of the negative number $−a$ is equal to the logarithm of the positive number $+a$. Thus, the position of M . Bernoulli implies that $l−a = l+a$. M. Leibniz gave occasion to this declaration of M. Bernoulli when he argued, in Letter CXC of the Exchange, that the ratio of $+1$ to $−1$, or of $−1$ to $+1$, was imaginary, because the logarithm of the measure of that ratio, that is to say the logarithm of $−1$, which is the exponent of that ratio, was imaginary. To this, M. Bernoulli declared, in Letter CXCIII, that he was not of this opinion, and that he believed in fact that the logarithms of negative numbers were not only real, but also equal to the logarithms of the same numbers, taken positively. M. Bernoulli supported his position with the following proofs.

[...]

POSITION OF M. LEIBNIZ

M. Leibniz holds that the logarithms of all negative numbers, and even more so those of imaginary numbers, are imaginary; thus, since $l−a = la + l−1$, he holds that $l−1$ is an imaginary quantity. I have already remarked that M. Leibniz held that the ratio of +1 to $−1$ or of $−1$ to $+1$ is imaginary, since the logarithm of that ratio or $l−1$ is imaginary. We see, of course, that all the objections which were made against the system of M. Bernoulli serve to strengthen this position, and that the reasons advanced to support M. Bernoulli's position must be contrary to those of M. Leibniz. Nevertheless, one can bring forward particular proofs to confirm M. Leibniz's position, which will be the subject of my examination which follows.

PROOF 1. Having observed that the logarithm of the number $1 + x$ is equal to the sum of this series $$l(1 + x) = x − \frac{1}{2}x^2+ \frac{1}{3}x^3-\frac{1}{4}x^4+\frac{1}{5}x^5-\frac{1}{6}x^6+ecc. $$ from which we see to begin with that if $x = 0$, it follows that $l1 = 0$, now to obtain the logarithm of $−1$ we must set $x = −2$, whence we obtain $$l−1 = -2 − \frac{1}{2}\cdot 4+ \frac{1}{3}\cdot 8-\frac{1}{4}\cdot 16+\frac{1}{5}\cdot 32-\frac{1}{6}\cdot 64-ecc.$$ Now, there is no doubt that the sum of this divergent series could not be $= 0$; thus, it is certain that $l−1$ is not $= 0$. The logarithm of $−1$ will thus be imaginary, since it is also clear that it could not be real, i.e., positive or negative.

[...]

It would thusappear that M. Leibniz’s view is better-founded, since it is not contrary to the discovery of M. Bernoulli, that is $$l\sqrt{}−1 = \frac{1}{2}\pi\sqrt{}−1,$$ since M. Leibniz holds that the logarithm of $−1$, and even more so that of $\sqrt{}−1$, is imaginary. But, in adopting M. Leibniz's position, we plunge into the aforementioned difficulties and contradictions. For, if $l−1$ were imaginary, its double, i.e. the logarithm of $(−1)^2 = +1$, would also be imaginary, which doesn't accord with the first principle of the theory of logarithms, in virtue of which we suppose that $l+1 = 0$.

[...]

SOLUTION OF THE PRECEDING DIFFICULTIES

It must first be stated that if the idea which Messrs. Leibniz and Bernoulli have attached to the term ‘logarithm’, and which all Mathematicianshave had up to now, were perfectly correct, it would be absolutely impossible to rescue the theory of logarithms from the contradictions which I have been propounding. Now, the idea of logarithms having been derived from an origin of which we have a perfect understanding, how is it possible that it could be defective? When we say that the logarithm of a given number is the exponent of the power of a certain number taken arbitrarily, which becomes equal to the given number, it appears that nothing is lacking to the correctness of that idea. And that is perfectly true; but we generally put that idea together with a condition which does not suit it at all: that is, we ordinarily suppose, almost without noticing it, that to each number there corresponds only one logarithm; now, with only a little consideration, we will find that all the difficulties and contradictions by which the theory of logarithms appears to be embarrassed, persist only to the extent that we suppose that to each number there corresponds only one logarithm.

The problem doesn't seem to be related to some integrals, but rather, as one can see also in the following, to infinite series. Moreover, from Leibniz and Euler consideration, it seems to me that logarithms of negative quantities arise as a natural extension of "real" logarithms (in Leibniz words: "the logarithm of $−1$ [...] is the exponent of the ratio of $+1$ to $−1$, or of $−1$ to $+1$" see above), provided that they are "in accord with the [...] principle[s] of the theory of logarithms".

If you can read French I also suggest this modern paper (Euler, d'Alembert et la controverse sur les logarithmes), ad also this 1759 original paper (Réflexions sur les quantités imaginaires) of Daviet de Foncenex, expecially at page 126.

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