Question

Which equation represents the magnitude of an

Answer 1

Answer:

$M=\log{(100S)}$

Step-by-step explanation:

The Richter scale, the standard measure of earthquake intensity, is a logarithmic scale, specifically logarithmic base 10. This means that every time you go up 1 on the Richter scale, you get an earthquake that's 10 times as powerful (a 2.0 is 10x stronger than a 1.0, a 3.0 is 10x stronger than a 2.0, etc.).

How do we compare two earthquake's intensities then? As a measure of raw intensity, let's call a "standard earthquake" S. What's the magnitude of this earthquake? The magnitude is whatever power of 10 S corresponds to; to write this relationship as an equation, we can say $10^m=S$, which we can rewrite in logarithmic form as $m=\log{S}$.

We're looking for the magnitude M of an earthquake 100 times larger than S, so reflect this, we can simply replace S with 100S, giving us the equation

$M=\log{(100S)}$.

To check to see if this equation is right, let's say we have an earthquake measuring a 3.0 on the Richter scale, so $3=\log{S}$. Since taking 100 times some intensity is the same as taking 10 times that intensity twice, we'd expect that more intense earthquake to be a 5.0. We can expand the equation $M=\log{(100S)}$ using the product rule for logarithms to get the equation

$M=\log{(100S)}=\log{100}+\log{S}$

And using the fact that $\log{100}=2$ and our assumption that $\log{S}=3$, we see that $M=2+3=5$ as we wanted.