Download power

symbolic notation  a  kz
Example - if k = 1500 years/m calculate sediment age at
depths of 1m, 2m and 5.3m. Repeat for k =3000 years/m
1m
Age = 1500 years
2m
Age = 3000 years
5.3m
Age = 7950 years
For k = 3000years/m
Age = 3000 years
Age = 6000 years
Age = 15900 years
The geologist’s use of math often turns out to be a
periodic and necessary endeavor. As time goes by you
may find yourself scratching your head pondering oncemastered concepts that you suddenly find a need for.
This is often the fate of basic power rules.
Evaluate the following
xa / xb =
xa+b
xa-b
(xa)b =
xab
xaxb =
Question 1.2a Simplify and where possible evaluate the
following expressions i) 32 x 34
ii) (42)2+2
iii) gi . gk
iv) D1.5. D2
Exponential notation is a useful way to represent
really big numbers in a small space and also for
making rapid computations involving large numbers for example,
the mass of the earth is
5970000000000000000000000 kg
the mass of the moon is 73500000000000000000000 kg
Express the mass of the earth in terms of the lunar mass.
Can you write the following numbers in
exponential notation (powers of 10)?
Density of the earth’s core is 11000 kg/m3
The volume of the earth’s crust is 10000000000000000000 m3
The mass of the earth’s crust is 28000000000000000000000kg
Differences in the acceleration of gravity on the earth’s
surface (and elsewhere) are often reported in milligals. 1
milligal =10-5 meters/second2.
Express the acceleration due to gravity of the earth (9.8m/s2)
in milligals.
The North Atlantic Ocean is getting wider at the average
rate vs of 4 x 10-2 m/y and has width w of approximately 5
x 106 meters.
1. Write an expression giving the age, A, of the North
Atlantic in terms of vs and w.
2. Evaluate your expression. When did the North Atlantic
begin to form?
Here is another polynomial which again attempts to fit the
near-surface 100km. Notice that this 4th order equation
(redline plotted in graph) has three bends or turns.
T  1.289 x10 11 d 4  1.99 x10 7 x3  0.00113 x 2  3.054 x  394 .41
th
4 Polynomial
5000
4000
T
3000
2000
1000
0
0
1000
2000
3000
4000
Depth (km)
5000
6000
7000
In sections 2.5 and 2.6 Waltham reviews negative
and fractional powers. The graph below illustrates
the set of curves that result as the exponent p in
y  ax p  a0
Is varied from 2 to -2 in -0.25 steps,
and a0 equals 0. Note that the
negative powers rise quickly up
along the y axis for values of x less
than 1 and that y rises quickly with
increasing x for p greater than 1.
Power Laws
1200
X
-2
1000
800
Y
2 4
600
What is 0.012 ?
400
2
What is 0.01 2 ?
X
-1.75
X
200
2
X
0
0
1
2
3
x
4
5
1.75
Power Laws - A power law relationship relevant to
geology describes the variations of ocean floor depth as a
function of distance from a spreading ridge (x).
d  ax1 / 2  d 0
Ocean Floor Depth
0
1
Spreading Ridge
2
D (km)
3
4
5
0
200
400
600
800
1000
X (km)
What physical process do you think might be responsible for this
pattern of seafloor subsidence away from the spreading ridges?
  0.6 x 2 z
This equation assumes that the initial porosity (0.6)
decreases by 1/2 from one kilometer of depth to the next.
Thus the porosity () at 1 kilometer is 2-1 or 1/2 that at the
surface (i.e. 0.3), (2)=1/2 of (1)=0.15 (I.e. =0.6 x 2-2 or
1/4th of the initial porosity of 0.6.
Equations of the type
y  ab cx
Are referred to as allometric growth laws or exponential
functions.
In the equation   0.6 x 2 z
y  ab cx
a=?
b=?
c=?
The constant b is referred to as a base.
Recall relationships like y=10a.
a is the power to which the base
10 is raised in order to get y.
y  10 x
By definition, we also say that x is the log of y,
and can write
 
log y  log 10 x  x
So the powers of the base are logs. “log” can be thought of
as an operator like x and  which yields a certain result.
Unless otherwise noted, the operator “log” is assumed to
represent log base 10. So when asked what is
log y, where y  45
We assume that we are asking for x such that
10 x  45
Sometimes you will see specific reference to
the base and the question is written as
log 10 y, where y  45
leaves no room for doubt that we are
log 10 y
specifically interested in the log for a base of 10.
One of the confusing things about logarithms is the word
itself. What does it mean? You might read log10 y to say ”What is the power that 10 must be raised to to get y?”
How about this operator? pow10  y
The power of base 10 that yields () y
log 10 y  1.653
N/year
537.03
389.04
218.77
134.89
91.20
46.77
25.70
16.21
8.12
4.67
2.63
0.81
0.66
2.08
1.65
1.09
0.39
0.23
0.15
0.12
0.08
0.04
0.03
Observational data for earthquake
magnitude (m) and frequency (N,
number of earthquakes per year)
600
Number of earthquakes per year
m
5.25
5.46
5.7
5.91
6.1
6.39
6.6
6.79
7.07
7.26
7.47
7.7
7.92
7.25
7.48
7.7
8.11
8.38
8.59
8.79
9.07
9.27
9.47
500
400
300
200
100
0
5
6
7
8
9
10
Richter Magnitude
What would this plot look like if we
plotted the log of N versus m?
The Gutenberg-Richter Relation
log N  bm  c
Number of earthquakes per year
1000
100
10
-b is the slope and
c is the intercept.
1
0.1
0.01
5
6
7
8
Richter Magnitude
9
10
In general,
log base (some number) 
log10 (number)
log10 base
or
log b a 
log10 (a)
log10 b
Try the following on your own
log 3 7 
log10 (7)
?
log10 3
log 8 8
log 7 21
log 4 7