primes and logarithms [theory] [closed]

Closed. This question is off-topic. It is not currently accepting answers.

---

Want to improve this question? Update the question so it's on-topic for开发者_如何学JAVA Stack Overflow.

Closed 12 years ago.

Improve this question

I know this is not a mathematical forum but given the bright minds that participate here, i am sure that this question is of interest nevertheless. How would you develop and explain the following statement:

"we can convert the product of a set of primes into a sum of the logarithms of the primes by applying logarithms to both parts of this conjecture"

---

log(a * b) = log(a) + log(b)

---

thanks for that OrangeDog and John!

re benefit of introducing logs, OrangeDog is right indeed. It is specific to an exercise from an MIT OpenCourse class. Here's the full details:

There is a cute result from number theory that states that for sufficiently large n the product of the primes less than n is less than or equal to e^n and that as n grows, this becomes a tight bound (that is, the ratio of the product of the primes to e^n gets close to 1 as n grows).

Computing a product of a large number of prime numbers can result in a very large number, which can potentially cause problems with our computation. [note: this is what John was referring to] So we can convert the product of a set of primes into a sum of the logarithms of the primes by applying logarithms to both parts of this conjecture. In this case, the conjecture above reduces to the claim that the sum of the logarithms of all the primes less than n is less than n, and that as n grows, the ratio of this sum to n gets close to 1.

EDIT

given these statements i am, however, unsure about how to apply them i.e.

how do we go from here:

2 x 3 x 5 <= e^7

to

"applying logarithms to both parts of this conjecture."

EDIT 2

got it...

2 x 3 x 5 <= e^7

knowing that logarithms are the opposite of powers we can say:

log(2x3x5) <= 7

which is also the same as:

log(2)+log(3)+log(5) <= 7

this only starts to show its "value" when n (in this case 7) gets larger i.e. the 1000th prime or higher