The Richter scale: From two readings, the relative strength and vice versa

In class last week, we learned how to find out how much stronger one quake is compared to another given the two Richter scale readings. For example, on June 28, the strongest quake in the U.S. was a 3.4 in Oklahoma, while the strongest in North America was a 5.4 in Niltepec, Mexico. What is the difference in levels of energy? Here are the steps.

Step 1: Subtract little from big. In our case, 5.4 - 3.4 = 2.0.
Step 2: multiply difference by 1.5. 2.0 * 1.5 = 3.0.
Step 3: Raise 10 to the power of the answer from Step 2: 10^3.0 = 1,000. The Mexican quake was 1,000 times stronger than the Oklahoma quake.

Let's ask the question in the opposite direction. Let's say we have a reading for a quake and we know another quake was x times stronger. Again, it will be a three step process, but now we will take the inverse of our three steps above in reverse order. Let's say we have a quake 350 times stronger than the one in Niltepec.  Here are our steps.

Step 1: Take the log of the strength multiplier. Log is the inverse of raising 10 to a power, just like addition is the inverse of subtraction and division is the inverse of multiplication. log(350) = 2.544...,
Step 2: divide the answer from Step 1 by 1.5 and round this answer to the nearest tenth. 2.544/1.5 = 1.696..., which rounds to 1.7. We round to the nearest tenth because the Richter scale rounds to the nearest tenth.
Step 3: Add the answer from Step 2 to the Richter reading we know. In this case, it would be 5.4+1.7 = 7.1, the reading of the stronger quake. If instead we were told a quake was 350 times weaker than Niltepec, it would be 5.4-1.7 = 3.7

Here are some practice questions. The answers are in the comments.

1. 16 times stronger than a 6.1
2. 250 times stronger than a 6.7
3. 8 times weaker than a 5.8

Logarithmic scales

Monday in class, there will be a handout concerning logarithms and their practical uses, notably in the Richter scale which measures earthquakes and the bel system which measures sound. Besides the handout, here are a few more practice problems.

Practice with logarithmic scales - Richter and decibel.

a) There was a 7.6 earthquake in Mindanao in the Philippines and a 5.7 aftershock three days later. Rounded to the nearest multiple of 10, how much stronger was the bigger quake?

b) An earthquake registering 5.7 is recorded one morning, and in the afternoon, another quake 27 times stronger is felt. Give the Richter reading of the second stronger quake. (Nearest tenth.)

c) What is the decibel reading of a sound that has 7 times more energy than a 75 decibel reading. (Nearest decibel.)

d) One singer is measured at 64 dB. Eight singers at the same level would have eight times more energy. To the nearest decibel, what is the decibel level of the eight singer chorus?

e)  How much more energy is there in a reading of 75 dB compared to 64 dB.  Round to the nearest whole number.

Answers in the comments.