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GG 313 Lecture 20 November 3, 2005 The Exponential Function and Logs

GG 313 Lecture 20 November 3, 2005 The Exponential Function and Logs. This lecture was originally going to be on multiple regression, but on reading Paul’s note I decided to talk about the exponential function instead.

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GG 313 Lecture 20 November 3, 2005 The Exponential Function and Logs

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  1. GG 313 Lecture 20 November 3, 2005 The Exponential Function and Logs

  2. This lecture was originally going to be on multiple regression, but on reading Paul’s note I decided to talk about the exponential function instead. I’ll talk for about 15 minutes, then show a webcast of a lecture downloaded yesterday on the exponential function, population, and energy. One of the greatest problems we face is getting people to understand that the concept of “sustainable growth” is unrealistic, and understanding the basic math and science as to why.

  3. The exponential function has the form: note that exp(0)=1 exp(1)=e exp(x+y)=exp(x)*exp(y) exp(x+iy)=ex*(cos(y)+i*sin(y)) exp(x)=cosh(x)+sinh(x) d(ex)/dx= ex d(loge(x))/dx=1/x

  4. exp(z) can be expanded in the Maclaurin series: If z=k*t= a constant times time, then exp(z) represents an exponential growth or exponential decay, depending on whether k is positive or negative: k= +0.7 k= -0.7

  5. Both exponential growth and exponential decay are important in earth sciences. Exponential decay extremely important in radioactive dating, and exponential growth is the subject of the video that follows. Note what happens if we take the log of exp(z): loge(exp(z))=z, which is no surprise, since this function DEFINES the log. If we let y=loge(x) , then ey=x The log TRANSFORMS multiplication into addition: Log(a * b)= log(a)+log(b) This is the principal behind the slide rule.

  6. If we plot y=exp(kt) on a log scale, the result is a straight line where log(y)=kt k=+0.7 k=-0.7 y t t

  7. The time it takes for a value on a growth or decay curve to reach twice or half of its value is called the doubling time or the half life.

  8. The half life (or doubling time) is equal to Log(2)/k. The video will show an interesting fact: the doubling time for an exponential growth rate is close to 70/(% rate of growth). Thus a growth rate of 1% per time unit doubles the value in 70 time units, and a rate of 7% doubles the value in 10 time units. While listening, evaluate the integrity of the presenter. Do you think he’s telling the truth? Exaggerating? Are his statistics reasonable?

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