Download PPT20

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.
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.
The exponential function has the form:
exp(z)  e , where
z is any number and e = 2.718...
z
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
exp(z) can be expanded in the Maclaurin series:

n
2
3
4
z
z z
z
exp( z)    1 z     ....
2 6 24
n0 n!
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
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.
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
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.
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?