![]() ![]() It means roughly "10x difference" but just sounds cooler than "1 digit larger". Taking log(500,000) we get 5.7, add 1 for the extra digit, and we can say "500,000 is a 6.7 figure number". With logarithms a ".5" means halfway in terms of multiplication, i.e the square root ($9^.5$ means the square root of 9 - 3 is halfway in terms of multiplication because it's 1 to 3 and 3 to 9). In our heads, 6.5 means "halfway" between 6 and 7 figures, but that's an adder's mindset. It gives a rough sense of scale without jumping into details.īonus question: How would you describe 500,000? Saying "6 figure" is misleading because 6-figures often implies something closer to 100,000. Talking about "6" instead of "One hundred thousand" is the essence of logarithms. Logarithms count the number of multiplications added on, so starting with 1 (a single digit) we add 5 more digits ($10^5$) and 100,000 get a 6-figure result. Adding a digit means "multiplying by 10", i.e. how many powers of 10 they have (are they in the tens, hundreds, thousands, ten-thousands, etc.). ![]() We're describing numbers in terms of their digits, i.e. Time for the meat: let's see where logarithms show up! When dealing with a series of multiplications, logarithms help "count" them, just like addition counts for us when effects are added. With the natural log, each step is "e" (2.71828.) times more. ![]() Logarithms describe changes in terms of multiplication: in the examples above, each step is 10x bigger. power of 80 = $10^80$ = number of molecules in the universeĪ 0 to 80 scale took us from a single item to the number of things in the universe.power of 23 = $10^23$ = number of molecules in a dozen grams of carbon.This smaller scale (0 to 100) is much easier to grasp: The trick to overcoming "huge number blindness" is to write numbers in terms of "inputs" (i.e. It's the difference between an American vacation year and the entirety of human civilization. Millions and trillions are "really big" even though a million seconds is 12 days and a trillion seconds is 30,000 years. Logarithms put numbers on a human-friendly scale. 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?"). 1000 is 10 which grew by itself for 3 time periods ($10 * 10 * 10$).100 is 10 which grew by itself for 2 time periods ($10 * 10$).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.īy the way, the notion of "cause and effect" is nuanced. 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. Logarithms find the cause for an effect, i.e the input for some outputĪ common "effect" is seeing something grow, like going from \$100 to \$150 in 5 years. Don't look for the literal symbols! When was the last time you wrote a division sign? When was the last time you chopped up some food? Ok, ok, we get it: what are logarithms about? Finding "math in the real world" means encountering ideas in life and seeing how they could be written with notation. Math expresses concepts with notation like "ln" or "log". Also, can you imagine a world without zinc?" "Scientists care about logs, and you should too. Most attempts at Math In the Real World (TM) point out logarithms in some arcane formula, or pretend we're geologists fascinated by the Richter Scale. Surprised that logarithms are so common? Me too. And an interest rate is the logarithm of the growth in an investment. You're describing numbers in terms of their powers of 10, a logarithm. ![]()
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