base 10 log question

[jenzao]

Is there a way to calculate multi digit base 10 logs without a calc or tables?

eg.. what is ? log (base 10) of 1381.42?? (without a calulator)??

[jenzao]

(with caveat that i memorized log 1 - 9)

[lurflurf]

Of coursethat is possible, the first people who made tables did not have calculators.
since
log(ab)=log(a)+log(b)
numbers should be decomposed multiplicatively.
One method is to factor into primes
log(1381.42)=-2+log(2)+2log(17)+log(239)
This shows a problem logs of many primes will be needed
another method would involve writing the number as a product of factors with known logs and factors near 1 since when x is small
log(1+x)~log(e)[x-x^2/2+x^3/3-x^4/4+...]
say one knew log(138176)
then
log(1381.42)=-2+log(138176)-log(1+1/4064)

[HallsofIvy]

What I would do is this:
First, log(1381.42)= log(0.138142 x104)= 4+ log(0.138142).

Now, use the 'Taylor's Polynomial' expansion:
for 0< x< 1, log(x)= x+ (1/2)x2+ (1/3)x^3+ ...+ (1/n)x^n.