We can solve this problem using an exponential equation and logarithms!
Step 1: Identify What We Know
Magnitude of the Indonesia earthquake = 9.1
The Indonesia earthquake was 74,900 times as strong as the Texas earthquake.
Every increase of 1.0 in magnitude corresponds to an earthquake that is 30 times stronger, according to the problem.
Let x = magnitude of the Texas earthquake.
Step 2: Write an Exponential Equation
Because each increase of 1.0 represents a factor of 30, the relationship is:
30^(9.1 - x) = 74,900
Here, (9.1 - x) represents the difference in magnitude between the two earthquakes.
Step 3: Solve Using Logarithms
Take the logarithm of both sides:
log(30^(9.1 - x)) = log(74,900)
Use the logarithm power rule:
(9.1 - x)log(30) = log(74,900)
Divide both sides by log(30):
9.1 - x = log(74,900) / log(30)
Calculate:
9.1 - x ≈ 3.30
Solve for x:
x ≈ 9.1 - 3.30
x ≈ 5.8
Final Answer:
The magnitude of the Texas earthquake was approximately 5.8.
How Does the Solution Help?
The logarithm tells us how many magnitude units separate the two earthquakes.
Since the Indonesia earthquake was approximately 3.3 magnitude units stronger, we subtract 3.3 from 9.1 to find the Texas earthquake's magnitude.
Remember:
Exponential equations describe repeated multiplication, while logarithms help us solve for unknown exponents.
In this problem:
30^(9.1 - x) = 74,900
And:
x = 9.1 - log(74,900)/log(30)
Answer: Approximately 5.8 on the magnitude scale.
Note: This solution uses the problem's stated approximation of 30 times stronger per magnitude unit.
