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Just by hand, only by hand, nothing but by hand? (Posted on 2004-07-26) Difficulty: 4 of 5
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 log10 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?

See The Solution Submitted by Federico Kereki    
Rating: 4.1667 (6 votes)

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Solution One Solution (first part) | Comment 3 of 7 |

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.


  Posted by DJ on 2004-07-26 14:34:32
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