How I can find the logarithm of 55 in base 10 using a pen not a calculator or other tool?

Question

I want to know the full procedure how to solve it?

Answer 1

10<55<100

log 10 <log55<log 100

1 < log55 < 2

Hence

log 55=1+0,xxxxx

Answer 2

you know the formula

ln (1+x) = x - (x^2/2) + (x^3/3) - (x^4/4) + .........

replacing x with -x, we have,

ln (1-x) = -x - (x^2/2) - (x^3/3) - ........

subtracting the two series, we get

ln {(1+x)/(1-x)} = 2 (x + x^3/3 + x^5/5 + ......... )

putting (1+x) / (1-x) = z,

           x = (z-1) / (z+1),

we have

ln (z) = 2 * Σ [1/(2n+1)]*[(z+1)/(z-1)]^n.

this formula works for all z>1. but if you want to calculate logs from it, its going to be very tedious.

you'd better stick to the log tables.

Answer 3

Use logtables. They are not "tools"