Math Is Fun Forum

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,476

Linear Interpolation

Logarithms are a very useful tool in calculations.

By remembering the log to base 10 values of just three numbers, the approximate log value of any number can be found.

All you have to remember is (all approximate values)
log(10)2=0.3010
log(10)3=0.4771
log(10)7=0.8451

log(10)4 would be 2log(10)2
log(10)5 would be 1 - log(10)2
log(10)6 would be log(10)2+log(10)3
log(10)8 would be 3log(10)2
log(10)10 would be 2log(10)3
log(10)10 is 1.

Illustration : To find the log(10)23,
we know log 20 is 1.3010,
log 25 is 2log(10)5, log(10)5 is 0.6990
Therefore, log(10)25 would be 1.3980
23 is 3/5 between 20 and 25.
Find the difference between log(10)25 and log(10)20. It is 0.0970
3/5 is 6/10, therefore, 6/10 of 0.0970 is 6*0.0097
that is 0.0582.
Add this to log(10)20.
You get 1.3592. This would be approximately log(10)23.
The scientific calculator gives the value as 1.3617.
Although there's a 0.0025, the percentage difference is only 0.18%.
This works much better for higher numbers.
Try finding the log(10)value of 82, you can calculate the log(10) values of 81 and 84 with what you remember. log(10)82 would be 1/4 of the difference added to log(10)81.
I found this method helpful.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Linear Interpolation

Hey uh, common log 82 is 1/2.9639 of the way between log81 and log84, which is close to 1/3.
Probably just a typo.

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igloo myrtilles fourmis

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,476

Re: Linear Interpolation

Thanks for correcting me, John,
yes, that was a typographic error.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.