This link has been bookmarked by 39 people . It was first bookmarked on 14 Jan 2008, by Chris D'Iorio.
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22 Jul 11
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The natural log gives you the time needed to reach a certain level of growth.
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02 Jan 11
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- e^x is the amount of continuous growth after a certain amount of time.
- Natural Log (ln) is the amount of time needed to reach a certain level of continuous growth
e and the Natural Log are twins:
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The number e is about continuous growth. As we saw last time, e^x lets us merge rate and time: 3 years at 100% growth is the same as 1 year at 300% growth, when continuously compounded.
We can take any combination of rate and time (50% for 4 years) and convert the rate to 100% for convenience (giving us 100% for 2 years). By converting to a rate of 100%, we only have time to think about:
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- How much growth do I get after after x units of time (and 100% continuous growth)
- For example: after 3 time periods I have e^3 = 20.08 times the amount of “stuff”.
ntuitively, e^x means:
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e^x is a scaling factor, showing us how much growth we’d get after x units of time.
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- e^x lets us plug in time and get growth.
- ln(x) lets us plug in growth and get the time it would take.
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The natural log gives us the time needed to hit our desired growth
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What is ln(1)? Intuitively, the question is: How long do I wait to get 1x my current amount?
Zero. Zip. Nada. You’re already at 1x your current amount! It doesn’t take any time to grow from 1 to 1.
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ln(negative number) = undefined
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I hope the strange math of logarithms is starting to make sense: multiplication of growth becomes addition of time, division of growth becomes subtraction of time. Don’t memorize the rules, understand them.
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Now the question is easy: How long to double at 100% interest? ln(2) = .693
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It takes .693 units of time (years, in this case) to double your money with continuous
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23 Dec 10
Frank Noschese@k8nowak How about this: http://bit.ly/fpiZFw ? Not great, but possibly useful...
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14 Sep 10
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11 Aug 10
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25 Apr 10
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- e^x is the amount of continuous growth after a certain amount of time.
- Natural Log (ln) is the amount of time needed to reach a certain level of continuous growth
e and the Natural Log are twins:
-
e^x lets us merge rate and time
-
- How much growth do I get after after x units of time (and 100% continuous growth)
- For example: after 3 time periods I have e^3 = 20.08 times the amount of “stuff”.
Intuitively, e^x means:
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- e^3 is 20.08. After 3 units of time, we end up with 20.08 times what we started with.
- ln(20.08) is about 3. If we want growth of 20.08, we’d wait 3 units of time (again, assuming a 100% continuous growth rate).
For example:
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e^x is a scaling factor, showing us how much growth we’d get after x units of time.
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- e^x lets us plug in time and get growth.
- ln(x) lets us plug in growth and get the time it would take.
Now what does this inverse or opposite stuff mean?
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What is ln(1)? Intuitively, the question is: How long do I wait to get 1x my current amount?
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ln(1) = 0
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ln(.5) = – ln(2) = -.693
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ln(negative number) = undefined
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Time to grow 4x = ln(4) = Time to double and double again = ln(2) + ln(2)
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ln(a*b) = ln(a) + ln(b)
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ln(5/3) = ln(5) – ln(3)
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ln(a/b) = ln(a) – ln(b)
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- e^x = growth
- e^3.4 = 30
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- 3.4 years at 100% = 3.4 * 1.0 = 3.4
- 1.7 years at 200% = 1.7 * 2.0 = 3.4
- 6.8 years at 50% = 6.8 * 0.5 = 3.4
- 68 years at 5% = 68 * .05 = 3.4
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- rate * time = .693
- time = .693/rate
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rate * time = .693
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Simple. As long as rate * time = .693 , we’ll double our money
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rate * time = .693
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time to double = 69.3/rate, where rate is assumed to be in percent.
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time to double = 72/rate
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- time to triple = 110 / rate
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The math robot says: Because they are defined to be inverse functions, clearly
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- (e) = 1
- The intuitive human: ln(e) is the amount of time it takes to get “e” units of growth (about 2.718). But e is the amount of growth after 1 unit of time, so ln(e) = 1.
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10 Apr 10
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01 Jul 09
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22 Apr 09
mel aclaroThe natural log gives you the time needed to reach a certain level of growth.
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The natural log gives you the time needed to reach a certain level of growth.
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- e^x is the amount of continuous growth after a certain amount of time.
- Natural Log (ln) is the amount of time needed to reach a certain level of continuous growth
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E is about growth
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Natural Log is about time
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- e^x lets us plug in time and get growth.
- ln(x) lets us plug in growth and get the time it would take.
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The natural log gives us the time needed to hit our desired growth.
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29 Jan 09
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12 Jan 09
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09 Jan 09
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07 Aug 08
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05 Aug 08
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17 Jun 08
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03 May 08
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23 Apr 08
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16 Mar 08
Jay DuggerGiven how the natural log is described in math books, there’s little “natural” about it: it’s defined as the inverse of e^x, a strange enough exponent already.
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28 Nov 07
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05 Oct 07
Alan DeanAfter understanding the exponential function our next target is the natural logarithm.
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