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

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.

Other versions of this theorem give

• 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).

As a final example, suppose we pick an integer at random from the interval [1,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)

• The constant e and its computation