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‘It has been known for over a century that as you progress up through the natural numbers, the average gap between one prime number p and the next is the natural logarithm of p.’
‘Euler asserts that the sum of the harmonic series equals the natural logarithm of infinity plus a quantity that is nearly a constant.’
‘According to the prime number theorem, the average gap length up to a certain prime equals the value of the natural logarithm of that prime.’
‘Benjamin can also handle magic squares, natural logarithms, cube roots, and much more.’
‘Variables were transformed with natural logarithms or square roots to normalize residuals.’
‘Due to a skewed distribution of the summed scale, the natural logarithm was used in the analysis.’
‘In his spare time, between 1844 and 1847, he calculated the natural logarithms of the first 1005000 numbers to 7 decimal places.’
‘Having arrived at the above equation, we can use the exponential function and the natural logarithm to solve it.’
‘Normally, when mathematicians talk about logarithms, they mean so-called natural logarithms, which are logarithms to base e.’
‘All variables used in this study were transformed to natural logarithms before the correlation analyses were performed.’