The Richter scale assigns a magnitude number to quantify the energy released by an earthquake. The magnitude of an earthquake was defined by Charles Richter to be $M =$ log$\frac { I } { S }$ where $I$ is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken $100$ km from the epicenter of the earthquake) and $S$ is the intensity of a “standard earthquake” (whose amplitude is $1$ micron $= 10^{-4}$ cm). For example, an earthquake measuring $3.6$ on the Richter scale is $10^{0.2}$ or approximately $1.6$ times as intense as an earthquake measuring $3.4$ on the Richter scale.

1. How many times stronger is an earthquake that measures $6.5$ on the Richter scale than an earthquake that measures $5.5$?

   Answer (a):

   $10$ times

2. How many times stronger is an earthquake that measures $5.1$ than an earthquake that measures $4.3$? Give your answer both as a power of $10$ and as a decimal. Make sure your answer has the same level of precision as the Richter measurements.

   Answer (b):

   $10^{0.8} ≈ 6.3$

3. What is the magnitude of an earthquake that is half as strong as an earthquake measuring $6.2$ on the Richter scale?

   Step 1 (c):

   Calculate the energy released by a Richter measurement of $6.2$. Then divide in half.

   Step 2 (c):

   $10^?=10^\frac{6.2}{2}$

   Hint (c):

   You may want to use logs or guess and check to find the answer.