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# Logarithmic Scales - PowerPoint PPT Presentation

Logarithmic Scales. Connecting qualitative observations with quantitative measurements. Logarithmic Scales. “A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself.” --- From the Wikipedia article on Logarithmic scales.

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### Logarithmic Scales

Connecting qualitative observations with quantitative measurements.

• “A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself.” --- From the Wikipedia article on Logarithmic scales.

• Question: Why would anyone want to measure the logarithm of quantity?

There are two very good reasons.

Logarithmic scale of the Universe

----XKCD

XKCD is my very favorite on-line comic strip.

OK… that’s cute, but what does it have to do with logarithms?

Suppose we consider the sequence . . .

What happens if the we the logarithm (base 10) of each term? We get . . .

If we start with a sequence in which the ratio from one term to the next is constant and we take logarithms, then the resulting sequence will have terms that are evenly spaced.

Suppose we consider the sequence . . .

What happens if the we the logarithm (base 10) of each term? We get . . .

More generally, if we start with a very spread out sequence that gets more and more spread out as we go along, taking logarithms, compresses the spread . . . This is how the XKCD cartoon allows us to take in the whole universe at a glance.

• In music---frequency of vibration vs. our perception of pitch. Our ears hear equalratios in frequency as equalchangesin pitch.

• For instance, concert A is 440 Hz in frequency. The A one octave below concert A is 220 Hz. The A one octave above concert A is 880 Hz. A change in pitch on 1 octave corresponds to a doubling of the frequency.

### Measuring Earthquakes

More on logarithmic scales and human perception

The severity of an earthquake can be expressed in terms of both intensity and magnitude.

• Intensity

• based on observed effects of ground shaking on people, buildings, and natural features.

• It varies from place to place within the disturbed region depending on the location of the observer with respect to the earthquake epicenter.

• Magnitude

• related to the amount of seismic energy released at the hypocenter of the earthquake.

• It is based on the amplitude of the earthquake waves recorded on instruments which have a common calibration.

• The magnitude of an earthquake is thus represented by a single, instrumentally determined value.

• Poarid (Italian)– 1627; first attempt to classify earthquakes by intensity in 1627; four levels of intensity use to describe effects of earthquakes experienced at different towns.

• More attempts made in 18th and 19th centuries.

• Early 1900’s Mercalli’s scale; 12 levels of intensity.

• Modified version (1931) published by Wood & Neumann. Known as the Modified Mercalli Intensity Scale of 1931, it has become the standard used by the United States engineering seismology community (incl. National Geophysical Data Center and the U.S. Geological Survey.)

• The ModifiedMercalli intensity scale is used for measuring the intensity of an earthquake.

• Quantifies the effects of an earthquake on the Earth's surface, humans, objects of nature, and man-made structures.

• 12 levels, with 1 denoting a weak earthquake and 12 one that causes almost complete destruction. (See handout)

• USGS is the Federal agency responsible for collecting earthquake intensity data.

• Used “Earthquake Report” questionnaire and field investigators to analyze earthquake damage in serious earthquakes.

• Questionnaires sent to postmasters, National Weather Service offices, military installations, etc. requesting a report of all effects of the earthquake in their area. Newspaper accounts, published scientific reports, and reports provided by seismology collaborators were also used.

• Expert observers may investigate and photograph the damage.

• In the end, a seismologist analyzes each report and assigns intensities on the basis of the effects at each town;

• Intensity (or isoseismal) maps are then constructed for earthquakes felt over large areas.

• Magnitude of most earthquakes is measured on the Richter scale, invented by Charles F. Richter in 1934. The Richter magnitude is calculated from the amplitude of the largest seismic wave recorded for the earthquake, no matter what type of wave was the strongest.

However, beware that such comparisons can be dicey, as damage depends on a lot more than the intensity of the Earthquake, such as the population density and the quality of construction materials used.

• Baseline: the “zero level earthquake” ---seismographic reading measures .001 millimeters at a distance of 100 km from the epicenter.

(Note: All seismographic readings are “adjusted” based on distance so that they represent the reading one would have gotten had one been 100 km from the epicenter.)

Set x0=.001 mm on seismograph---zero level

Suppose we have a seismograph reading of x mm. We are interested in the ratio

Now we define the Richter magnitude of the Earthquake by