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logarithmic functionThe function log x has two different standard meanings. In most high school and lower division college courses, log x is the common logarithm: the power to which ten must be raised to get x. In this sense, log 100 = 2.
However, in most upper division college courses, mathematical publications, and these pages, log x is the natural logarithm: the power to which
e = 2.7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274 2746639193 2003059921 8174135966 2904357290 ...must be raised to get x. (This is sometimes denoted ln x in lower level texts and on most calculators.) So here log 100 = 4.60517... and log10 x = (log x)/(log 10).
Why this is the "natural" base to use when dealing with prime numbers? The Prime Number Theorem states
- the number of primes ≤ x is asympototic to x/log x.
- the nth prime is about n log n,
- log(p#) ~ p (see p-primorial), and
- the average number of composites between the primes less than n is log n (see prime gaps).
Related pages (outside of this work)
Printed from the PrimePages <t5k.org> © Reginald McLean.