Logarithmic spiral - Wikipedia
This is due to the fact that this equation establish a relationship between the . ln (-1)=i*Pi. -->Mathematica: In:= Log[-1]. Out= \[ImaginaryI] \[Pi] and so on. Given 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 . This relationship makes sense when you think in terms of time to grow. God bless you!. More: “You, your pupils, your friends, God. Not a bad public . tables (or slide rules which are mechanized log tables) to do almost all of the world's scientific and.
What is ln 1? Intuitively, the question is: How long do I wait to get 1x my current amount? If we reverse it i.
Logarithm properties review
If we go backwards. In general, you can flip the fraction and take the negative: This means if we go back 1. Ok, how about the natural log of a negative number?
Logarithmic Multiplication is Mighty Fun How long does it take to grow 4x your current amount? Sure, we could just use ln 4. We can consider 4x growth as doubling taking ln 2 units of time and then doubling again taking another ln 2 units of time: Any growth number, like 20, can be considered 2x growth followed by 10x growth.
Or 4x growth followed by 5x growth. Or 3x growth followed by 6. This relationship makes sense when you think in terms of time to grow.
Why does ln(i) = (1/2pi)i? | Physics Forums
If we want to grow 30x, we can wait ln 30 all at once, or simply wait ln 3to triple, then wait ln 10to grow 10x again. The net effect is the same, so the net time should be the same too and it is.
Well, growing 5 times is ln 5. Suppose we want 30x growth: We can consider the equation to be: We can figure this out by letting the continuous growth run for a year: The year-over-year gain is 3.
From an instant-by-instant basis, a given part of the economy is growing by 3. In science and engineering, we prefer modeling behavior on an instantaneous basis. Do bacteria colonies replicate on clean human intervals, and do we wait around for an exact doubling? The growth factor is: Just for fun, how long until the bacteria doubles?
Imagine waiting for 1 to turn to 2: We know the rate was. Figuring out whether you want the input cause of growth or output result of growth is pretty straightforward. Imagine we have little workers who are building the final growth pattern see the article on exponents: Green at the end of the year.
But… that worker he was building Mr. Green starts working as well. Green first appears at the 6-month mark, he has a half-year to work same annual rate as Mr.
Blue and he builds Mr. Red ends up being half done, since Mr. Green only has 6 months.
Green showed up after 4 months? If workers begin growing immediately, we get the instant-by-instant curve defined by ex: We plug that rate into ex to find the final result, with all compounding included. Using Other Bases Switching to another type of logarithm base 10, base 2, etc. Each logarithm asks a question when seeing a change: What was the instantaneous rate followed by each worker?
How many doublings were required? How many 10x-ings were required? Over 30 years, the transistor counts on typical chips went from to 1 billion How would you analyze this?