How to calculate 3x7 by using logarithm?

Question

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 $3 \times 7$. So the helper asked Newton. After a quick thinking using logarithm, he got the result that $3 \times 7$ 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 $20<3\times7<22$? 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.

Answer 1

This story is very unlikely to be true, but anyway...

$$\log(3 \times 7) = \log(3) + \log(7)$$

Assuming we're using base-10 logarithms and that Newton has memorized some base-10 logs to two decimal places:

$$\eqalign{ 0.47 <& \log(3) < 0.48\cr 0.84 <& \log(7) < 0.85\cr 1.31 <& \log(3)+\log(7) < 1.33\cr}$$

Since $\log(20) < 1.31$ and $\log(22) > 1.34$, that gives you the answer.

Answer 2

It is from Shaw's play In Good King Charles's Golden Days.

MRS BASHAM. Oh, do look where youre going, Mr Newton. Someday youll walk into the river and drown yourself. I thought you were out at the university.

NEWTON. Now dont scold, Mrs Basham, dont scold. I forgot to go out. I thought of a way of making a calculation that has been puzzling me.

MRS BASHAM. And you have been sitting out there forgetting everything else since breakfast. However, since you have one of your calculating fits on I wonder would you mind doing a little sum for me to check the washing bill. How much is three times seven?

NEWTON. Three times seven? Oh, that is quite easy.

MRS BASHAM. I suppose it is to you, sir; but it beats me. At school I got as far as addition and subtraction; but I never could do multiplication or division.

NEWTON. Why, neither could I: I was too lazy. But they are quite unnecessary: addition and subtraction are quite sufficient. You add the logarithms of the numbers; and the antilogarithm of the sum of the two is the answer. Let me see: three times seven? The logarithm of three must be decimal four seven seven or thereabouts. The logarithm of seven is, say, decimal eight four five. That makes one decimal three two two, doesnt it? What's the antilogarithm of one decimal three two two? Well, it must be less than twentytwo and more than twenty. You will be safe if you put it down as--

Sally returns.

SALLY. Please, maam, Jack says it's twentyone.

NEWTON. Extraordinary! Here was I blundering over this simple problem for a whole minute; and this uneducated fish hawker solves it in a flash! He is a better mathematician than I.

Answer 3

I too have come across this story. It is in a book titled An Anthology of Anecdotes, where small biographies of scientists are presented with a view to show their human side; and also a balance between their achievements and their eccentricities. While it is clearly mentioned as Newton taking out his "Logarithmic Tables" in that book, I am yet to be able to find evidence (or lack thereof) regarding its authenticity.