
In the "olden days", a slide rule was a great way of estimating. For those of us born before 1960, it came naturally to us that log_{10}2 is about 0.3, and from this we estimated log_{10}4 as 0.6, log_{10}8 as 0.9, and log_{10}5 as 0.7. (I'll start dropping the subscript, 10, of log, because all logs are base 10 on this page.)
Now, log 50 is about 1.7, because every zero at the end of the number adds 1 to the log. 50 is very close to the square of 7, so log 7 is about 0.85.
And log 80 is about 1.9, and 80 is close to the fourth power of 3, so log 3 is about 1.9/4 = 0.475.
From the discussion, above, we see that:
log 2 ≈ .3 = 12/40
log 3 ≈ .475 = 19/40
log 4 ≈ .6 = 24/40
log 5 ≈ .7 = 28/40
log 6 ≈ .775 = 31/40
log 7 ≈ .85 = 34/40
log 8 ≈ .9 = 36/40
log 9 ≈ .95 = 38/40
Checking to see if other denominators yield close approximations to the base 10 logarithms of singledigit numbers, I noticed that 568 serves as a very good one.
Log 2 is very close to 171/568, so log 4 is close to 342/568, log 8 is close to
513/568, and log 5 = (568171)/568 = 397/568.
Log 1.5 is almost exactly 100/568, which makes it easy to calculate log 3 = 271/568, log 6 = 442/568,
and log 9=542/568.
That leaves log 7, which is amazingly close to 480/568.
log 2 ≈ 171/568
log 3 ≈ 271/568
log 4 ≈ 342/568
log 5 ≈ 397/568
log 6 ≈ 442/568
log 7 ≈ 480/568
log 8 ≈ 513/568
log 9 ≈ 542/568
A measure of what makes a "good denominator" is the average error of the numerators, when the logs of integers 2 through 9 are estimated by a fraction of this denominator. A timehonored way to find the "average error" is the "root mean square" method (RMS), in which the squares of the errors are averaged, and then the square root of the result is taken. If the errors of the numerators are randomly distributed between 0 and 0.5, then you would expect the RMS value to be close to the square root of average value of x^{2} between 0 and 0.5. This is the square root of twice the integral from 0 to 0.5 of x^{2}, or about 0.288675. So any denominators that give significantly smaller RMS errors are "good denominators". A list of successively better denominators is 1, 4, 7, 10, 13, 23, 40, 63, 176, 239, 329, 568, 10381, 49128, 60974, 281746, 342720, 7484108, 11452573, 18936681, 44390284, 55842857. (Sloane's A119256)
Karl's Calculus Tutor: Log base 10 tricks (to the 40th degree)
Word Problems, including percents, and the meaning of the words "of", "per", and "what", and much, much more.
The Expansions page gives methods for calculating values, such as Gauss' method for calculating pi.
The webmaster and author of this Math Help site is Graeme McRae.