aanpalendmw

2022-07-17

How to calculate $3×7$ by using logarithm?
This is a story about Newton I read once when I was a child. Now that book is lost and I can only tell you what I remember.
When Newton was young, he had been already famous in curiosity and smart. His family hired a helper. One day, she asked him to go to the market with her because she wasn't good at math. At the market, there was a problem that needed to calculate $3×7$. So the helper asked Newton. After a quick thinking using logarithm, he got the result that $3×7$ must larger than 20 and smaller than 22...
So, my question is how did he do that calculate? How to use logarithm to get the result $20<3×7<22$? Thank you so much.
The story countinues:
... and about to say the result. Before he could finish his math, a near by person had been listened to the conversation and jumped in: "3 times 7 is 21". "Wow, you are smarter than my Newton", said by the helper. "Indeed, you are smarter than Newton", Newton laughed away.

cindysnifflesuz

Expert

This story is very unlikely to be true, but anyway...
$\mathrm{log}\left(3×7\right)=\mathrm{log}\left(3\right)+\mathrm{log}\left(7\right)$
Assuming we're using base-10 logarithms and that Newton has memorized some base-10 logs to two decimal places:
$\begin{array}{rl}0.47<& \mathrm{log}\left(3\right)<0.48\\ 0.84<& \mathrm{log}\left(7\right)<0.85\\ 1.31<& \mathrm{log}\left(3\right)+\mathrm{log}\left(7\right)<1.33\end{array}$
Since $\mathrm{log}\left(20\right)<1.31$ and $\mathrm{log}\left(22\right)>1.34$, that gives you the answer.

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