Imagine you are living a lot of years ago, without calculators, slide rules, tables of logarithms, or any kind of tool but paper and pencil... how would you go about calculating log
10 of 2 -- or of any other number?
By the way, if your solution required calculating powers, or if you just wanted to check your solution, how would you calculate 10 to the 0.30103... power, or any other?
There are infinite series that can be used to find the value of a natural log. Then, log(2) = ln(2) / ln(10).
There are two series I know of that we can use:
ln(1+x) = x - x2/2 + x3/3 - x4/4 + x5/5 ...
ln[(1+x)/(1-x)] = 2×[x + x3/3 + x5/5 + x7/7 + ... ]
First, find ln(2):
ln(2) = ln [(4/3)/(2/3)] ln[(1+1/3)/(1-1/3)]
= 2×[1/3 + 1/(3×33) + 1/(5×35) + 1/(7×37) + 1/(9×39)]
= 2×[0.333333 + 0.012345 + 0.000823 + 0.000065 + 0.000006]
= 0.693146
These series converge much more quickly when x is small, so we won't compute ln(10) directly.
ln (5/4) = ln [(10/9)(8/9)] = ln [(1+1/9)/(1-1/9)]
= 2×[1/9 + 1/(3×93) + 1/(5×95)]
= 2×[0.111111 + 0.000457 + 0.000003]
= 2×[0.111570]
= 0.22314
ln(10) = ln(5/4) + ln(8) = ln(5/4) + 3 ln(2)
= 0.22314 + 3(0.693146)
= 0.22314 + 2.079438
= 2.302578
And finally:
log(2) = ln(2)/ln(10)
= 0.693146 / 2.302578
= 0.301030410261889
Which is accurate to 5 places, using only 7 long divisions and a few multiplications.
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Posted by DJ
on 2004-07-26 14:34:32 |
One Solution (first part)
