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オイラー【⇐「」での比較記事】を以下で考えてみます。具体的には「単眼の巨人(サイクロプス)と呼ばれたEuler in Switzerlandは自然対数を定式化しましたが、その仮定を具体的に教えて下さい。」と記載し以下を得ます。⇒

Euler, a Swiss mathematician, did indeed contribute significantly to the development of the natural logarithm. His work on logarithms, including the natural logarithm, played a crucial role in the advancement of mathematics.

The natural logarithm, denoted as ln(x), is a logarithm with a base of the mathematical constant e, which is approximately equal to 2.71828. Euler’s contribution to the natural logarithm involved formulating a series expansion for ln(x).

Euler’s supposition, known as Euler’s series or Euler’s formula for the natural logarithm, states that any positive real number x can be expressed as the following infinite series:

ln(x) = (x – 1) – (1/2)(x – 1)^2 + (1/3)(x – 1)^3 – (1/4)(x – 1)^4 + …

This series allows for the calculation of the natural logarithm of a given number by approximating it as the sum of infinitely many terms. By truncating the series after a certain number of terms, one can obtain an increasingly accurate approximation of ln(x).

Euler’s work on the natural logarithm and his series expansion provided a significant breakthrough in understanding logarithmic functions, paving the way for further developments in calculus, analysis, and various branches of mathematics.