Seismology- word problem

Question

An earthquake of magnitude 7.7 occurred in 2001 in Gujarat, India. It was about 4900 times as strong as the greatest earthquake ever to hit Pennsylvania. What is the magnitude of the Pennsylvania earthquake?

Answer 1

Logarithmic Scales

Earthquakes are measured on the Richter Scale, or something like it. These scales are logarithmic, which means that every time the magnitude 𝑀 increases by 1...

𝑀₂ = 1 + 𝑀₁

...the amplitude 𝐴 increases by a factor of 10.

𝐴₂ = 10 · 𝐴₁

This amplitude is a physical measurement, which corresponds to the actual strength of the earthquake. More generally, if we increase the magnitude by 𝑥...

𝑀₂ = 𝑥 + 𝑀₁

...then the amplitude 𝐴 increases by a factor of 10^𝑥.

𝐴₂ = 10^𝑥 · 𝐴₁

We can state this formula using logarithms, where we abbreviate 𝗅𝗈𝗀₁₀ as simply 𝗅𝗈𝗀.

𝑀₂ − 𝑀₁ = 𝗅𝗈𝗀(𝐴₂ ⁄ 𝐴₁)

Problem

We are asked for the magnitude 𝑀_𝑃 of the Pennsylvanian earthquake. We are told the magnitude 𝑀_𝐺 of the Gujarati earthquake...

𝑀_𝐺 = 7.7

...and that it was 4,900 times stronger than the Pennsylvanian earthquake. In other words, the Gujarati amplitude 𝐴_𝐺 was 4,900 times larger than the Pennsylvanian amplitude 𝐴_𝑃.

𝐴_𝐺 = 4900 · 𝐴_𝑃

Solution

We can take our formula for magnitude, and solve for 𝑀_𝑃 using algebra.

𝑀_𝐺 − 𝑀_𝑃 = 𝗅𝗈𝗀(𝐴_𝐺 ⁄ 𝐴_𝑃)

𝑀_𝑃 = 𝑀_𝐺 − 𝗅𝗈𝗀(𝐴_𝐺 ⁄ 𝐴_𝑃)

The crucial trick is to substitute 4900 · 𝐴_𝑃 in place of 𝐴_𝐺 which is unknown.

𝑀_𝑃 = 𝑀_𝐺 − 𝗅𝗈𝗀(4900 · 𝐴_𝑃 ⁄ 𝐴_𝑃)

= 𝑀_𝐺 − 𝗅𝗈𝗀(4900)

Our equation now contains values we do know, so we can “plug in” the 7.7 for 𝑀_𝐺 and calculate the result.

𝑀_𝑃 = 7.7 − 𝗅𝗈𝗀(4900) = 4.00980391997...

So the Pennsylvanian earthquake had a magnitude of roughly 4.0