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 1000 S .
Simplify the fraction: M = lo g 1000 .
Express 1000 as a power of 10: M = lo g 1 0 3 .
Apply the logarithm property to find the magnitude: M = 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 told that the earthquake is 1,000 times more intense than a standard earthquake, which means l = 1000 S . We need to find the magnitude M of this earthquake.
Substitution Substitute l = 1000 S into the formula: M = lo g S 1000 S .
Simplification Simplify the expression by canceling out S : M = lo g 1000 .
Expressing 1000 as a power of 10 Since 1000 = 1 0 3 , we have M = lo g 1 0 3 .
Applying Logarithmic Property 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. Since 3 is an integer, rounding to the nearest tenth gives 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 relates to its intensity compared to a standard earthquake. In real life, understanding earthquake magnitudes helps in assessing potential damage and preparing for seismic events. For example, an earthquake of magnitude 6.0 is ten times stronger than an earthquake of magnitude 5.0.
The magnitude of an earthquake that is 1,000 times more intense than a standard earthquake is 3.0. This calculation uses the formula M = lo g S l and results in M = lo g 1000 = 3 . Therefore, rounded to the nearest tenth, the magnitude is 3.0.
;
