Logarithms and Earthquakes
Earthquakes are measured by - the magnitude (Richter Scale)
- the intensity (compared to a standard earthquake)
Intensity (I) is measured by how many more times intense the earthquake is when compared to a
standard earthquake.
Example: A standard earthquake’s intensity is denoted by I0 (“I naught”). In December
2004, an earthquake in the Indian Ocean was 1 billion times more intense than
a standard earthquake. So, if we call the intensity of the 2004 earthquake I,
then the intensity of the 2004 earthquake can be written: I = 1,000,000,000I0.
I0 could be considered the unit of the intensity.
𝑰
Magnitude (R) is measured on what is called the Richter Scale where 𝑹 = 𝐥𝐨𝐠⁡(𝑰 ) .
𝟎
Example: What is the magnitude of an earthquake whose intensity is 72,000 times I0?
Solution: Since we are asked to find the magnitude, we need to find R. To use
the formula for R, we need to write the intensity I as 72,000I0 and
then substitute this value for I into the equation for the magnitude.
72,000𝐼0
𝑅 = log (
𝐼0
)
𝑅 = log⁡(72,000)
𝑹 ≅ 𝟒. 𝟗
The magnitude of this earthquake was 4.9.
Example: What is the intensity of an 8.3 magnitude earthquake?
Solution: Let R = 8.3 in the formula that relates intensity to magnitude.
𝐼
8.3 = log⁡( )
𝐼0
To solve for I we have to first convert this log equation to an
exponential equation:
𝐼
𝐼
𝐼0
𝐼0
8.3 = log⁡( ) ⇔ ⁡108.3 = ⁡
Now we can solve for I, by multiplying both sides by I0:
𝐼
⁡×⁡ 𝐼0
𝐼0
108.3 𝐼0 = ⁡𝐼 or ⁡⁡⁡𝑰 = ⁡𝟏𝟗𝟗, 𝟓𝟐𝟔, 𝟐𝟑𝟐⁡𝑰𝟎
𝐼0 ⁡× 108.3 = ⁡
The intensity of an 8.3 magnitude earthquake is 199,526,232 I0.
In general, the intensity can be found using the equation, I = 10RI0.