The magnitude, [tex]$M$[/tex], of an earthquake is defined to be [tex]$M=\log \frac{l}{S}$[/tex], where [tex]$l$[/tex] is the intensity of the earthquake (measured by the amplitude of the seismograph wave) and [tex]$S$[/tex] is the intensity of a "standard" earthquake, which is barely detectable. What is the magnitude of an earthquake that is 1,000 times more intense than a standard earthquake? Use a calculator. Round your answer to the nearest tenth.
Answer (2)
Substitute the given intensity l = 1000 S into the magnitude formula M = lo g S l .
Simplify the expression to M = lo g S 1000 S = lo g 1000 .
Use the fact that 1000 = 1 0 3 to get M = lo g 1 0 3 .
Apply the logarithmic property lo g 1 0 x = x to find the magnitude: 3.0 .
Explanation
Understanding the Problem We are given the formula for the magnitude of an earthquake: M = lo g S l , where l is the intensity of the earthquake and S is the intensity of a standard earthquake. We are also given that the earthquake is 1,000 times more intense than a standard earthquake, so l = 1000 S . Our goal is to find the magnitude M of this earthquake.
Substituting the Given Value Substitute l = 1000 S into the magnitude formula: M = lo g S 1000 S .
Simplifying the Expression Simplify the expression: M = lo g 1000 .
Expressing 1000 as a Power of 10 Since 1000 = 1 0 3 , we have M = lo g 1 0 3 .
Using Logarithmic Properties Use the property of logarithms that lo g 1 0 x = x to find M = 3 .
Rounding the Answer Round the answer to the nearest tenth, which is 3.0.
Final Answer Therefore, the magnitude of the earthquake is 3.0.
Examples
Earthquakes release energy that can be measured using the Richter scale, which is a logarithmic scale. This problem demonstrates how the magnitude of an earthquake is calculated based on its intensity relative to a standard earthquake. Understanding earthquake magnitudes helps in assessing the potential damage and impact on communities. For example, an earthquake with a magnitude of 6.0 is ten times stronger than an earthquake with a magnitude of 5.0.
The magnitude of an earthquake that is 1,000 times more intense than a standard earthquake is calculated to be 3.0. Using the formula M = lo g S l and substituting l = 1000 S , we find that M = lo g 1000 = 3.0 . Therefore, the answer is rounded to 3.0.
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