Uh oh, what was that answer again?
The number of 12-letter combinations from the phrase "BIOLOGY FOR FUN AND PROFIT" was 5.67 × 1013 . Both Google and your calculator will give you something like that, rather than a civilized number like 56693912432456.
Why? Well, 56693912432453 might be too big to fit on your calculator's display. Or it might not, but you have to admit it's a difficult number to read. Is it billions? Trillions? Quadrillions?
Instead, scientists (and calculators and browsers) use scientific notation:
1013 is a one with thirteen zeros, or 10,000,000,000,000. Which is 10 trillion, incidentally.
So 5.67 × 1013 is about 57 trillion.
What is the log of 5.67 × 1013 (approximately)?
- I need a hint ... : Remember that the log tells you how many "spaces" the number takes up.
- ...another hint ... : The 1013 tells you the number takes up 13 spaces.
- ...another hint ... : 5.67 × 1013 is about halfway between 1013 and 1014.
I think I have the answer: A little more than 13.5. The actual answer is 13.75.
And you can see, there is a nice relationship between scientific notation and the log scale.
Notice, finally, that 5.67 × 1013 does NOT include all the digits of the exact answer. Even though the exact answer is valid in this case, it would be a pain to write it all out, and frankly, whether there are 56693912432453 words or 56693912432459 words or even 56693999999999 words doesn't make much difference. When you're talking about TRILLIONS, it's like the old joke, a million here, a million there, maybe someday you're talking real money.
So in summary, there are 4 ways we could express this number:
- about 57 trillion
- 5.67 × 1013
- 13.75 on a log scale
We'll use the scientific notation method because (as you'll see) it will make our later calculations easier.
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