I still have a cheap Casio I use from time to time. Using it to calculate logarithms using only multiplication to the 10th power and the occasional divide by 10 was a calming thing to do over lunch in the gear shop without Internet access.

Start with X. If 10 or more add 1 to the log10 for each decimal If smaller than 1, subtract 1 each time you multiply by 10.writw that down plus a decimal point

Now the fun part, the fraction of the logarithm

Hit multiply 9 times then equal, to get the 10th power on the Casio

Each time you divide by 10 to get it back into range between 1 and 10, add 1 to the fraction

Repeat until you've got enough digits

PS: it also works for binary or any other integer base.