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Calculating in your head is gradually becoming a lost art, and indeed there is a generation of children coming up these days, most of whom need a calculator to multiply six times nine. But there is nothing like the ability to make rough calculations in your head to both impress your friends and give you a sense of mastery over the numerical world. And as an added bonus, it endows you some measure of protection against those who would dupe you out of your hard-earned money.

As you learned in algebra, the log 10 function was once used extensively to carry out multiplication problems. A table of 5-place logarithms was, perhaps, the most used reference book in any math library. But in your head and without a printed log table, you can carry out crude logarithm calculations based solely on a few key multiplication facts. These facts lead you to good working approximations of the logs base 10 of the counting numbers from 1 through 10 .

If you are a computer programmer, you probably already know that 210 = 1024 . And even if you're not, that fact is still true. And 1024 is very near to 1000 = 103 . This, together with the rules of logarithms leads to

log 10 (210) = 10 log 10 (2) ≈ log 10 (1000) = 3

≈

10

where the symbol,, means "approximately equal to." And dividingout of that you get

log 10 (2) ≈ 3 10

Now indulge me for a minute for preferring to express this fraction as

log 10 (2) ≈ 12 40

My reasons will be evident momentarily.

So what about log 10 (3) ? Well, we have 34 = 81 , and that's very nearly 80 . So what? you might ask. Well, 80 = 10 × 23 , and we already know log 10 (10) and an approximation to log 10 (2) . So we have

log 10 (34) = 4 log 10 (3) ≈ log 10 (80) = log 10 (10) + 3 log 10 (2)

log 10 (2)

Using the approximation for, that becomes

4 log 10 (3) ≈ 40 + 3 × 40 12 = 40 76 40

Dividing out the 4, we get

log 10 (3) ≈ 19 40

log 10 (4)

4

22

4

2

To get an approximation to, simply observe thatis. So the log ofis twice the log of

log 10 (4) ≈ 24 40

log 10 (5)

5

10

For, observe thatis half of. So

log 10 (5) = log 10 (10) - log 10 (2) ≈ 40 - 40 12 = 40 28 40

6

2

3

2

3

For, we know it is the product ofand. So the approximation to its log will be the sum of the approximations to the logs ofand. In other words

log 10 (6) ≈ 12 + 40 19 = 40 31 40

7

72 = 49

50

log 10 (5)

log 10 (50)

1

Which brings us to. Hmmm. Observe that, and that's very nearly. Well we already have an approximation for, so we can findsimply by addingto it.

log 10 (72) = 2 log 10 (7) ≈ 28 + 40 40 = 40 68 40

2

Divide through byand you've got it.

log 10 (7) ≈ 34 40

8

9

Bothandare easy.

log 10 (8) = 3 log 10 (2) ≈ 36 40

and

log 10 (9) = 2 log 10 (3) ≈ 38 40

10

1

10

40

0, 12, 19, 24, 28, 31, 34, 36, 38, and 40

Hands On Project: Making a Slide Rule Here's something you can try using some cardboard (file-folder material works too), a compass, and a protractor. This project is thanks to Chuck Fete, who teaches high school math. Cut two circles, one larger than the other. Mount them concentrically with a pin or thumbtack through the common center. Now mark out angles corresponding to the logs base 10 of the numbers, 1 through 9 . Since there are 360 degrees in a circle, each fortieth corresponds to 9 degrees. Use the estimates for the logs in fortieths given above to figure out the angle for each number. An image of the final product is shown to the right. By rotating the inner circle, you can multiply by various numbers. The diagram shows three seconds each of multiplying by 1 through 6 . Observe how the angles line up between the circles for the different multipliers. See if you can interpret the alignments for yourself based upon the multiplication tables you undoubtedly memorized years ago. Teachers: For more information on this project

4

f(x) = 10 log 10 (x)

There you have it -- approximations to the logs baseof the numbers fromto, inths arerespectively. If you don't believe me, divide each of them byon a calculator and see how close they come to values listed asin figure 6.3-1 in the main text.

Now let's do a quickie calculation using these. Suppose you wanted to know 10! . That's the product of the numbers from 1 to 10 . And we know that the log of the product is the sum of the logs. So add these numbers up (which is an exercise many folks can do in their heads) and you get 262 . When you divide by 40 , you get 6 with a remainder of 22 . The 6 tells us that 10! is somewhere between one million and ten million (that is, it's in the 106 decade, and 106 will be the characteristic). Now what about the remainder? It tells us the mantissa. Find where the remainder fits in the list of numbers above. It falls three fifths of the way between 19 and 24 , which are the approximations for log 10 (3) and log 10 (4) respectively. Three fifths of the way between 3 and 4 is 3.6 . So the answer is 3.6 million. How accurate is that? Well, in reality 10! = 3628800 . Our approximation is off by less than 1% .



Here's another quickie table that's useful for doing still more calculations.

x 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 40 log 10 (x) 0.17 0.34 0.51 0.68 0.85 1.0 1.2 1.3 1.5 x 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 40 log 10 (x) 1.7 3.2 4.6 5.8 7.0 8.2 9.2 10.2 11.2 table 6.3-2: 40 log 10 Approximations

log 10 (1.06)

25

25

40

25/40

log 10 (4)

log 10 (5)

I had to cheat and use a calculator to make this, so for it to be useful, you either have to memorize it or make a cheater-card and keep it in your wallet. Then, for example, if somebody wanted to know what $1000 would be left for 25 years at 6% interest, you could take the approximation for, multiply it byto get, then remember that's inths and thatis one fourth of the way between the approximation we had forand. Then you'd know that you would end up with about $4250 (the actual is $4292).

With all this in mind, try multiplying all the approximations by 6 , so that each represents 240 log 10 (x) , and see if you can make sense of the rule of 72, a financial rule of thumb which states that to find out how many years it takes your money to double, divide the annual interest rate (in percentage points) into 72.

And by the way, how do you think I knew that the logs base 10 of the integers 2 through 9 are irrational? Pick any natural number that is not a power of 10 . Call it n . If log 10 (n) were rational, then there would be two natural numbers, p and q , such that p/q = log 10 (n) . But then

p = q log 10 (n) = log 10 (nq)

log 10 (x)

10x

10p = nq

n

p

q

n

10

Sinceis theof, it must also be true thatRemember that, andare all counting numbers. See if you can make an argument based upon prime factorization of the two sides of the above equation to show why this equation is impossible unlessis a power of

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