Quandaries and Queries


my name is Amanda, and I'm going into my senior year of high school.  I will be taking AP calculus, and my teacher gave us some homework over the summer.  However, there are two things that I do not understand how to do.  The first is, she wants us to be able to generate a unit circle by hand using 30, 60 and 90 degree triangles.  I have used the unit circle in trigonometry, however I was never taught how to draw it.  Secondly, I need to know how to do natural logarithms without a calculator.  I was not taught how to do this, and the worksheet I was given only showed me how to complete them using a calculator. 
Thank you so much for any help that you might be able to offer


Hi Amanda,

This looks like a very tough class. I was taught how to draw a circle using compasses, pennies or glasses, but never with triangles. Also, I know how to compute the natural logarithm using tables, a slide rule or a calculator, but without these tools, the best I can do is approximate them. (And even Napier http://turnbull.mcs.st-and.ac.uk/history/Mathematicians/Napier.html,
the guy who devised logarithms, had no easy way to compute them; he made up tables and eventually invented the slide rule.)

To approximate natural logarithms, you can make a small table as follows: the base e is about 2.7, so that ln(2.7) is approximately1. Then, e e is approximately 7.3, so that ln(7.3) is approximately2. Then, e e e is approximately 19.7, so that ln(19.7) is approximately 3, and so on.

In this way, you make up a list 2.7, 7.3, 19.7, 53.2, ... of numbers whose
natural logarithms are approximately 1, 2, 3, 4, ... For numbers that fall between
these values, you need to interpolate: 10 is between 7.3 and 19.7, so
ln(10) should be between 2 and 3. Perhaps 2.2 is a good estimate;
try it out and see how it works.

There is another way to approximate natural logarithms if you are allowed to use one of these solar powered calculators with a square root button but no ln button:

  1. enter the number whose logarithm you want to calculate (say 19.7)
  2. press the square root button ten times
  3. subtract 1
  4. multiply by 1024.

On my calculator, I get 2.9849... when I perform these operations. This is quite good since it is supposed to be an approximation of ln(19.7) which is about 3. You can try it with 2.7 and 7.3 to convince yourself that it works well enough. Then one day in your second or third calculus course you will learn why this works.



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