General Math (Logarithm)

Using Logarithms in the Real World

 

A common "effect" is seeing something grow, like going from $100 to $150 in 5 years. How did this happen? We're not sure, but the logarithm finds a possible cause: A continuous return of ln(150/100) / 5 = 8.1% would account for that change. It might not be the actual cause (did all the growth happen in the final year?), but it's a smooth average we can compare to other changes.
By the way, the notion of "cause and effect" is nuanced. Why is 1000 bigger than 100?
  • 100 is 10 which grew by itself for 2 time periods (10 · 10)
  • 1000 is 10 which grew by itself for 3 time periods (10 · 10 · 10)
We can think of numbers as outputs (1000 is "1000 outputs") and inputs ("How many times does 10 need to grow to make those outputs?"). So, Why is this useful?

Logarithms put numbers on a human-friendly scale.
Large numbers break our brains. Millions and trillions are "really big" even though a million seconds is 12 days and a trillion seconds is 30,000 years. It's the difference between an American vacation year and the entirety of human civilization.
The trick to overcoming "huge number blindness" is to write numbers in terms of "inputs" (i.e. their power base 10). This smaller scale (0 to 100) is much easier to grasp:
  • power of 0 = 100 = 1 (single item)
  • power of 1 = 101 = 10
  • power of 3 = 103 = thousand
  • power of 6 = 106 = million
  • power of 9 = 109 = billion
  • power of 12 = 1012 = trillion
  • power of 23 = 1023 = number of molecules in a dozen grams of carbon
  • power of 80 = 1080 = number of molecules in the universe
A 0 to 80 scale took us from a single item to the number of things in the universe. Not too shabby.
Logarithms count multiplication as steps
Logarithms describe changes in terms of multiplication: in the examples above, each step is 10x bigger. With the natural log, each step is "e" (2.71828...) times more.
When dealing with a series of multiplications, logarithms help "count" them, just like addition counts for us when effects are added.

 

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