How was the concept of $\ln(x)$ found before the man knows that it is the area under hyperbola or it is related to the power of $e$ (base of logarithm). How did Napier compute the value of $e$ or $\ln(x)$ before the concept of power of numbers was used in math?
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1See Florian Cajori's 7-part series of papers History of the exponential and logarithmic concepts in American Mathematical Monthly, Volume 20 (1913), pp. 5-14, 35-47, 75-84, 107-117, 148-151, 173-182, 205-210. – Dave L Renfro May 15 '19 at 07:22
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1Before the invention of the logarithms, long multiplications used to be performed by the prosthaphaeresis method, based on the product-to-sum trigonometric formulas. In fact, these were already logarithms, but in the imaginary domain ($e^{ia}e^{ib}=e^{i(a+b)}$). – May 15 '19 at 07:30
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Powers of numbers were known long before Napier. – May 15 '19 at 07:37
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See Brian Rice & Enrique González-Velasco & Alexander Corrigan, The Life and Works of John Napier (2017, Springer). – Mauro ALLEGRANZA May 15 '19 at 07:41
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If the concept of power of numbers were known before Napier,there were no use for it unless it is applied in logarithm – May 15 '19 at 07:47
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See also this: https://hsm.stackexchange.com/questions/42/which-came-first-the-natural-logarithm-or-the-base-of-the-natural-logarithm/47#47 – Alexandre Eremenko May 16 '19 at 12:55