How was the first log table put together?

Henry Briggs compiled the first table of base-$10$ logarithms in 1617, with the help of John Napier. My question is: how did he calculate these logarithms? How were logarithms calculated back then?

I've found these pages to be fairly useful, but they don't seem to say much in the way of what I'm looking for. Any answers or useful references would be appreciated.

At MAA you may find 'Logarithms: The early history of a familiar function' while Napier's logarithms are described with care in Roegel's article 'Napier's ideal construction of the logarithms' (rather less nice than the usual ones since using $10^7$ as reference!). A shorter description was given by Lexa in 'Remembering John Napier and His Logarithms' and should provide a good starting point.

Napier's work itself appears in 'A Description of the Admirable Table of Logarithms' : Edward Wright's $1616$ translation of Napier's Latin book.
A book from $1915$ named 'Napier tercentenary memorial volume' is proposed by archive.org.

You can find the details in e: The Story of a Number. The basic idea is that square roots are easy to calculate. If you want for example $\log_{10}2$ (the number such $2=10^{\log_{10}2}$): $$10^{0.25} = 10^{1/4} = 1.778...< 2 < 3.162... = 10^{1/2} = 10^{0.5},$$ i.e., $$0.25 < \log_{10}2 < 0.5$$ and multiplying/dividing for $10^{1/2^k}=\sqrt{\sqrt{\cdots\sqrt{10}}}$ you can get better approximations.

Also important: the successive square roots of 10 are calculated once and can be used many times.