Calculate ln97 and log_(10)97 without calculator accurate up to 2 decimal places.

Francis Oliver

Francis Oliver

Answered question

2022-10-16

Calculate ln 97 and log 10 97 without calculator accurate up to 2 decimal places.
I have rote some value of logs of prime numbers up to 11
97 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

exalantaswo

Beginner2022-10-17Added 14 answers

You could write
2 ln ( 97 ) = ln ( 97 2 ) ln ( 96 98 )
to get higher precision, now your argument is off by 0.01% instead of 1% so the approximation will be close enough. log 10 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 ln ( 96 ), 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

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
97 × 5 = 485
This is very convenient since
485 1 = 484 = 4 × 11 2
and
485 + 1 = 486 = 6 × 81
Thus using the difference of two squares,
( 97 × 5 ) 2 1 = 484 × 486
and is therefore expressible in prime factors no higher than 11
Therefore we have
97 2 × 25 = 235225 235224 = 2 3 × 3 5 × 11 2
Therefore
ln 97 3 2 ln 2 + 5 2 ln 3 + ln 11 ln 5
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.

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