Save for primes under ten.................................either estimate of the log of pi......................overestimates them........it tends to generally.................................but after ten......10 to the one power........................the 1st several are all under the true number......

Look it.............there are 4 primes under ten.............................it is easier to see below................rounded up ........using MS Excel.................................after the 1st number........which is 10 to the one.....or just 10....................for the 1st couple of instances..................it tends to UNDERestimate...........the true number, from there(except maybe for farther up the number line)........it overestimates them....................these are rounded up...............so it is less than 25..............25 is actually the true number..............but my log of pi to 3.1 and 3.14.........the below.............respectively................there are 168 primes up to 1000...............10 to the 3rd.............again........UNDERestimate.................but at 10 to the 5.........it begins to OVERestimate them.......

9592.............is the true number of primes under 10 to the 5th.............(100,000).....................according to wiki..............

78498.............is the true number under a million...................10 to the 6th..................either log of pi...............OVERestimates that..............with over 80,000 primes...................

5		5
25		25
164		166
1,228		1,242
9,827		9,939
81,894		82,822
701,945		709,899
6,142,021		6,211,621
54,595,744		55,214,405
491,361,694		496,929,648
4,466,924,489		4,517,542,255