Exponential and Logarithm functions

jadewest · 2022-09-27 00:53:07

Hello,

I can't solve this exercise.

In an earthquake, a Seismic wave travels through the Earth layer, which gives out an energy that causes the earth to shake and also gives out low frequency acoustic energy. The instrument seismograph is based on a logarithmic scale, called the Richter Scale. Since it’s a base 10 scale each number increase on the scale indicates an intensity 10 times stronger than the previous number on the scale.

The Midlands of South Carolina have been experiencing a “swarm” of earthquakes in recent months. The strongest registered 3.7 on the Richter Scale. If the smallest recorded during the swarm was 2.7, what is the mathematical relationship between the two recorded earthquakes? Explain.

Thank you,
Jade

Jai Ganesh · 2022-09-27 02:49:38

Hi jadewest,

I may not help you directly, however, I can send two relevant links.

Seismic Waves

Introduction to Logarithms

Seismic magnitude scales are used to describe the overall strength or "size" of an earthquake. These are distinguished from seismic intensity scales that categorize the intensity or severity of ground shaking (quaking) caused by an earthquake at a given location. Magnitudes are usually determined from measurements of an earthquake's seismic waves as recorded on a seismogram. Magnitude scales vary on what aspect of the seismic waves are measured and how they are measured. Different magnitude scales are necessary because of differences in earthquakes, the information available, and the purposes for which the magnitudes are used.

The Earth's crust is stressed by tectonic forces. When this stress becomes great enough to rupture the crust, or to overcome the friction that prevents one block of crust from slipping past another, energy is released, some of it in the form of various kinds of seismic waves that cause ground-shaking, or quaking.

Magnitude is an estimate of the relative "size" or strength of an earthquake, and thus its potential for causing ground-shaking. It is "approximately related to the released seismic energy."

Intensity refers to the strength or force of shaking at a given location, and can be related to the peak ground velocity. With an isoseismal map of the observed intensities (see illustration) an earthquake's magnitude can be estimated from both the maximum intensity observed (usually but not always near the epicenter), and from the extent of the area where the earthquake was felt.

The intensity of local ground-shaking depends on several factors besides the magnitude of the earthquake, one of the most important being soil conditions. For instance, thick layers of soft soil (such as fill) can amplify seismic waves, often at a considerable distance from the source, while sedimentary basins will often resonate, increasing the duration of shaking. This is why, in the 1989 Loma Prieta earthquake, the Marina district of San Francisco was one of the most damaged areas, though it was nearly 100 km from the epicenter. Geological structures were also significant, such as where seismic waves passing under the south end of San Francisco Bay reflected off the base of the Earth's crust towards San Francisco and Oakland. A similar effect channeled seismic waves between the other major faults in the area.

Magnitude scales

Typical seismogram. The compressive P-waves (following the red lines) – essentially sound passing through rock – are the fastest seismic waves, and arrive first, typically in about 10 seconds for an earthquake around 50 km away. The sideways-shaking S-waves (following the green lines) arrive some seconds later, traveling a little over half the speed of the P-waves; the delay is a direct indication of the distance to the quake. S-waves may take an hour to reach a point 1000 km away. Both of these are body-waves, that pass directly through the earth's crust. Following the S-waves are various kinds of surface-waves – Love waves and Rayleigh waves – that travel only at the earth's surface. Surface waves are smaller for deep earthquakes, which have less interaction with the surface. For shallow earthquakes – less than roughly 60 km deep – the surface waves are stronger, and may last several minutes; these carry most of the energy of the quake, and cause the most severe damage.

An earthquake radiates energy in the form of different kinds of seismic waves, whose characteristics reflect the nature of both the rupture and the earth's crust the waves travel through. Determination of an earthquake's magnitude generally involves identifying specific kinds of these waves on a seismogram, and then measuring one or more characteristics of a wave, such as its timing, orientation, amplitude, frequency, or duration. Additional adjustments are made for distance, kind of crust, and the characteristics of the seismograph that recorded the seismogram.

The various magnitude scales represent different ways of deriving magnitude from such information as is available. All magnitude scales retain the logarithmic scale as devised by Charles Richter, and are adjusted so the mid-range approximately correlates with the original "Richter" scale.

Most magnitude scales are based on measurements of only part of an earthquake's seismic wave-train, and therefore are incomplete. This results in systematic underestimation of magnitude in certain cases, a condition called saturation.

zetafunc · 2022-09-27 07:33:44

Hi jadewest,

The key bit is here:

jadewest wrote:

each number increase on the scale indicates an intensity 10 times stronger than the previous number on the scale.

In other words, if you've got an earthquake A measuring 1 on the Richter scale and earthquake B measuring 2 on the Richter scale, earthquake B has an intensity 10 times stronger than earthquake A. Does that make sense?

What would happen if earthquake A measured 2 on the Richter scale and earthquake B measured 4 on the Richter scale -- what would the difference in intensity be then?

imcute · 2022-09-29 00:17:13

Ten.
Log(something)-Log(something)=3.7-2.7=Log(something)=1
so that last something is ten

Bob · 2022-09-30 19:37:26

hi imcute

Welcome to the forum.

I think zetafunc was trying to encourage Jade to work it out for herself. If you want her to undetstand (and I know she is keen to learn not just have the answers) then please develop your formula so the whole method becomes clear. Thanks.

Bob

Math Is Fun Forum

#1 2022-09-27 00:53:07

Exponential and Logarithm functions

#2 2022-09-27 02:49:38

Re: Exponential and Logarithm functions

#3 2022-09-27 07:33:44

Re: Exponential and Logarithm functions

#4 2022-09-29 00:17:13

Re: Exponential and Logarithm functions

#5 2022-09-30 19:37:26

Re: Exponential and Logarithm functions

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