log relationships, trig functions, earthquakes & computer lab tom.h.wilson...
TRANSCRIPT
Log relationships, trig functions, earthquakes &
computer [email protected]
Department of Geology and GeographyWest Virginia University
Morgantown, WV
LogarithmsThe allometric or exponential functions are in the form
cxaby and
cxay 10b and 10 are the bases. These are constants and we can define any other number in terms of these constants raised to a certain power.
xyei 10 .. Given any number y, we can express y as 10 raised to some power x
Thus, given y =100, we know that x must be equal to 2.
xy 10
By definition, we also say that x is the log of y, and can write
xy x 10loglogSo the powers of the base are logs. “log” can be thought of as an operator like x (multiplication) and which yields a certain result. Unless otherwise noted, the operator “log” is assumed to represent log base 10. So when asked what is
45y where,log yWe assume that we are asking for x such that
4510 x
Sometimes you will see specific reference to the base and the question is written as
45y where,log10 yy10log leaves no room for doubt that we are
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? -
ypow 10
Tom Wilson, Department of Geology and Geography
ypow 10
The power of base 10 that yields () y
653.1log10 y 1.65310 45
We’ve already worked with three bases: 2, 10 and e. Whatever the base, the logging operation is the same.
10.get to toraised bemust 5 power that theis what asks 10log5
? 10log5
How do we find these powers?
5log
10log 10log
10
105
431.1699.0
1 10log5
105 431.1 thus
In general, base
numberbase
10
10
log
)(log number) some(log
or
b
ab
10
10
log
)(log alog
Try the following on your own
?)3(log
)7(log 7log
10
103
8log8
21log7
7log4
10So log is often written as log, with no subscript
log10 is referred to as the common logarithm
ln. asten often writ is log e
2.079 ln8 8log e
thus
loge or ln is referred to as the natural logarithm. All other bases are usually specified by a subscript on the log, e.g.
etc. ,logor og 25l
Worksheet – pbs 16 & 17: sin(nx)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 45 90 135 180 225 270 315 360
sin()
See basics xlsx
Graphical sketch problem similar to problem 18
What approach could you use to graph this function?
Could get two points right away ….
What is x when y=0 &
What is y when x=0?
Approach
Finish up work on these in-class problem.Individually show your work.
Tom Wilson, Department of Geology and Geography
5 6 7 8 9 10
Richter Magnitude
0.01
0.1
1
10
100
1000
Num
ber
of e
arth
quak
es p
er y
ear
cbmN log
Recall discussions from last time on the Gutenberg-Richter Relation
-b is the slope and c is the intercept.
5 6 7 8 9 10
Richter Magnitude
0.01
0.1
1
10
100
1000
Num
ber
of e
arth
quak
es p
er y
ear
cbmN log
The Gutenberg-Richter Relation
m, the earthquake magnitude represents logarithmic differences in ground movements. For example - ground motion produced by a magnitude 5 earthquake is ten times the amplitude of ground motion produced by magnitude 4 earthquake.
The Richter magnitude scale determines the magnitude of shallow earthquakes from surface waves according to the following equation
3.3log66.1log10 T
Am
where T is the period in seconds, A the maximum amplitude of ground motion in m (10-6 meters) and is the epicentral distance in degrees between the earthquake and the observation point.
More logs!
Spend a few minutes in group discussion on today’s problems
See the basics.xls spreadsheet
Have a look at the basics.xlsx file
Some of the worksheets are interactive allowing you to get answers to specific questions. Plots are automatically adjusted to display the effect of changing variables and constants
Just be sure you can do it on your own!
Spend the remainder of the class working on Discussion group problems. The one below should be
handed in today.
Tom Wilson, Department of Geology and Geography
If you have extra time continue working on Discussion worksheet 2
What’s Due? In Summary!
Tom Wilson, Department of Geology and Geography
Due Today
Due next Thursday
Due Tuesday
• Look over problems 2.11 through 2.13
• Work on the earthquake frequency and porosity problem (due next time)
• Try and complete most of warm-up exercise Part 2 (it will be due next Thursday
•Continue your reading
Hand in discussion group problems (1) :sin(6x), logs, y = |2x+7| & f(x)=|3x+5|
Begin computer lab next time. Machines will be reformatted so we’ll wait till Tuesday to jump
in.
Tom Wilson, Department of Geology and Geography