Log relationships, trig functions, earthquakes & computer lab tom.h.wilson Department of Geology and Geography West Virginia University Morgantown, WV

Logarithms The allometric or exponential functions are in the form and b 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. 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.

By definition, we also say that x is the log of y, and can write So 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 We assume that we are asking for x such that

Sometimes you will see specific reference to the base and the question is written as 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 log 10 y to say - ”What is the power that 10 must be raised to to get y?” How about this operator? -

Tom Wilson, Department of Geology and Geography The power of base 10 that yields (  ) y

We’ve already worked with three bases: 2, 10 and e. Whatever the base, the logging operation is the same. How do we find these powers?

In general, or Try the following on your own

log 10 is referred to as the common logarithm thus log e or ln is referred to as the natural logarithm. All other bases are usually specified by a subscript on the log, e.g.

Worksheet – pbs 16 & 17: sin(nx) 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

Recall discussions from last time on the Gutenberg-Richter Relation -b is the slope and c is the intercept.

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 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