Calculate ln97 and log_(10)97 without calculator accurate up to 2 decimal places.
Francis Oliver
Answered question
2022-10-16
Calculate and without calculator accurate up to 2 decimal places. I have rote some value of logs of prime numbers up to is a little big. In case it would have been a multiple of smaller primes, I would have used the trick of logarithm identities . But I am confused how to do it, or will it be ok to approximate it to 96 or 98. I also don't know much calculus except differentiation and integration. I look for a short and simple way. I have studied maths up to 12th grade.
Answer & Explanation
exalantaswo
Beginner2022-10-17Added 14 answers
You could write
to get higher precision, now your argument is off by 0.01% instead of 1% so the approximation will be close enough. goes similiar of course. Edit: This would give accuracy to three decimal places and the fourth would probably be one off. If you instead would have chosen to approximate it by , you would have been one off in the second decimal and thus your answer would have been wrong. (I have computed the values with a calculator)
Ryder Ferguson
Beginner2022-10-18Added 3 answers
We can use the fact that the OP has a list of logarithms of prime numbers up to 11. Consider that
This is very convenient since
and
Thus using the difference of two squares,
and is therefore expressible in prime factors no higher than Therefore we have
Therefore
This approximation is correct to 5 decimal places BTW I think it is in the spirit of the question to use the information given by the questioner, and simple arithmetic, rather that resort to calculus and series expansions.