Ease multiplication with log 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.
 A: 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.
A: 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 $1.$
