The natural base, e, 2.718281828..., was selected because the derivatived/dx ex is equal to x. This simplifies many calculations, derivatives, integrals, etc.
Additional:
This base is used because there is a series (of terms) for powers of (e), therefore, powers of any value can be found just by plugging in values in the series.
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The natural logarithm (ln) is used when you have log base e
The logarithm of 1.5 is approximately 0.1760912591... Your logarithm is base 10, and the natural logarithm of 1.5 (base e), is approximately 0.4054651081... Example base: 8 Approximately: 0.1949875002...
Zero, in logs to base 10, base e, or any base.
The natural logarithm is calculated to base e, where e is Euler's constant. For any number, x loge(x) = log10(x)/log10(e)
The logarithm of 1 to the base 1 is indeterminate. The logarithm of a number x to the base a is a number y, such that ay = x. The most common base a is 10, or the natural base a is e (2.718281828...). It is invalid to think of logarithms base 1, because 1 to the power of anything is still 1.