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It is said that John Napier (1550-1617) and Joost Buergi (1552-1632) both were frustrated by the time spent multiplying numbers together. That's why they came up with the idea to replace multiplication by addition using logarithms.

As an example i found this: 

log(20) + log(50)  ≈  1.3 + 1.7 = 3
10*3 = 1000

However, how did they do those calculations? Were they realizing that it wasn't accurate enough?

log(20) = 1.30102999566

log(50) = 1.69897000434

Would be nice if someone could shed more light on that.

user1767754
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  • In the past, they used to create giant books of logarithms and trig function values. To find the log of $20$, you'd look the the value for $\log(2)$ in the table, then add $1$. Then, after doing the additions, you'd do a reverse lookup. – Thomas Andrews Jun 17 '17 at 00:20
  • When I was in high school (early 1980s) they still taught us how to use such tables, and how to interpolate values that were between two values in the table. – Thomas Andrews Jun 17 '17 at 00:21
  • A crude method to find logs might be something like: $2^{10} = 1024 \approx 10^3$, so $10 \log 2 \approx 3$ so $\log 2 \approx 0.3$. By taking higher powers of 2 you could get more and more accurate approximations. (Of course, I'm sure the actual methods used must have been more sophisticated than this.) – Daniel Schepler Jun 17 '17 at 00:24
  • Back before computers, when ships were navigated based on the sky, there were people aboard who were responsible for figuring out location based on the moon and stars. They used complex lookup tables for logarithms and trig functions, and also pre-computed locations for the moon in the sky. A great (young adult) book about this is "Carry On, Mr. Bowditch." – Thomas Andrews Jun 17 '17 at 00:26
  • Here's an online version of a book of logarithm values: https://archive.org/details/logarithmictable00joneuoft – Thomas Andrews Jun 17 '17 at 00:28
  • @ThomasAndrews thanks for that link! That's very exciting! – user1767754 Jun 17 '17 at 00:33
  • Each table is different in terms of numbers of digits. If you look at the Wikipedia page it has an picture of a 1617 example and a picture of a 20th century example. https://en.wikipedia.org/wiki/Mathematical_table#Tables_of_logarithms – Thomas Andrews Jun 17 '17 at 00:41

2 Answers2

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According to a Wikipedia page, Napier published the first logarithm table, using the natural log. There were 90 pages of values.

Biggs suggested using the standard log (base 10) to make it easier, because you only need to know the starting digits. He computed the values from $1$ to $1000$, published with Napier.

Even as late as the early 80s, I was taught to use tables to lookup up trigonometric function values, and interpolate values that were "between" table elements.

So, somebody had to do the computations to compute logarithms up front, but then everybody else had an easier time.

Here's and example of such a book.

Bowditch's American Practical Navigator, first published in 1802, was still required to be aboard all US vessels, last time I heard. It contained enough tables to allow fairly uneducated crew to compute navigation values with only lookups and the ability to add numbers. It included trig, logarithm, and lunar data (pre-computed lunar positions.)

There's a great young adult book, Carry On Mr Bowditch, about the writing of this book, and how many lives it saved.

You might also be interested in slide rules.

Thomas Andrews
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One pre-calculus method that was used to calculate logs was to take a number $x$ close to $1,$ say $x=1.000 001$ and by repeated squaring, find that $10$ is about $x^{2 302 585}$ and that $2$ is about $x^{698 971}$, from which $\log_{10}2$ is about $698 971/ 2 302 585.$.... After building a catalog of common logs of some small primes we can compute other logs by interpolation. E.g to compute $\log (23.5)$ we write it as $3\log 2 +\log 3 -\log (1-1/48),$ as it is less work to compute logs of numbers (like $1-1/48$) that are close to $1.$

DanielWainfleet
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