"The theory of natural ("hyperbolic") logarithms apparently first suggested itself to mathematicians engaged in the mensuration of spaces between the hyperbola and its asymptotes. About a quarter of a century later, in 1695, Edmund Halley discarded geometrical figures and published a remarkable article containing a purely arithmetical theory of logarithms. In this original and meritorious investigation he lays great stress upon what we now call the "modulus". By Napier's logarithms Halley understands those which give Briggs's logarithms when divided by 2.302 585 or when multiplied by 0.43429448. From this statement it appears that Halley considered Napier's logarithms to be identical with natural logarithms, and we must look upon him as one of the first (perhaps the first) to commit this error. That the two systems are not identical is shown by the following formula:\log_{N} x = 10^7 \log_{e} \frac{10^7}{x},During the eighteenth century this misunderstanding regarding the two systems does not appear to have been as wide-spread as it was later."
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Notation logN indicates "Napier" logarithm and loge, the natural logarithm.
https://en.wikiquote.org/wiki/History_of_logarithms
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