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# logarithmic function

The 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

must be raised to gete= 2.7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274 2746639193 2003059921 8174135966 2904357290 ...

*x*. (This is sometimes denoted ln

*x*in lower level texts and on most calculators.) So here log 100 = 4.60517... and log

_{10}

*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
*n*th 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).

*n*]. The probability that this random integer is prime, is about 1/log

*n*. (Technically, the probability is asympotic to 1/log

*n*as

*n*approaches infinity.)

**Related pages** (outside of this work)

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