Log relationships, trig functions … & computer lab

tom.h.wilsontom.wilson@mail.wvu.edu

Department of Geology and GeographyWest Virginia University

Morgantown, WV

Logarithms

The 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 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.

ypow 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? -

The power of base 10 that yields () y

653.1log10 y

What do you think? Are small earthquakes much more common than large ones?

Fortunately, the answer to this question is yes, but is there a relationship between the size of an earthquake and the number of such earthquakes?

5 6 7 8 9 10

Richter Magnitude

0

100

200

300

400

500

600

Num

ber

of e

arth

quak

es p

er y

ear

Observational data for earthquake magnitude (m) and frequency (N, number of earthquakes per year with magnitude greater than m)

What would this plot look like if we plotted the log of N versus m?

0.01

0.1

1

10

100

1000

Num

ber

of e

arth

quak

es p

er y

ear

5 6 7 8 9 10

Richter Magnitude

This looks like a linear relationship. Recall the

formula for a straight line?

bmxy

0.01

0.1

1

10

100

1000

Num

ber

of e

arth

quak

es p

er y

ear

5 6 7 8 9 10

Richter Magnitude

Where

Ny log

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

-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

is linear in a log-log format because m is earthquake magnitude and you have heard that an earthquake magnitude of 5, for example, represents ground motion whose amplitude is 10 times that associated with a magnitude 4 earthquake.

One of the most commonly used “Richter magnitude” scales 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.

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

?3log

)7(log 7log

10

103

8log8

21log7

7log4

subscript no with log, asten often writ is log that find You will 10

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

Spend a few minutes in group discussion on the two problems handed out Tuesday

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!

Worksheet – pb 15: 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()

Depth (cm)

0 500 1000 1500 2000 2500

AG

E(y

ears

BP

)

0

5000

10000

15000

North Sea Data (Lehman &Keigwan, 1992)

O18

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Lehman and Keigwin undertook one of the first deep sea studies to document changes in sea surface temperature associated with deglaciation

Computer lab…

Finish reading Chapters 1 and 2 (pages 1 through 38) of Waltham

Continue working on this lab for next class

Hand in the second set of warm-up questions