Question

the magnitude, \\(m\\), of an earthquake is defined to be \\(m = \log \frac{i}{s}\\), where \\(i\\) is the intensity of the earthquake (measured by the amplitude of the seismograph wave) and \\(s\\) is the intensity of a \standard\ earthquake, which is barely detectable. which equation represents the magnitude of an earthquake that is 10 times more intense than a standard earthquake?

\\(m = \log \frac{i}{10s}\\)
\\(m = \log(10s)\\)
\\(m = \log \frac{10s}{s}\\)
\\(m = \log \frac{10}{s}\\)

Explanation:

⚡ Using what you learned: Understanding Logarithmic Functions

Step 1: Identify the given formula and variables

The magnitude \(M\) of an earthquake is given by the formula:

$$ M = \log \frac{I}{S} $$

where:

• \(I\) is the intensity of the earthquake.
• \(S\) is the intensity of a standard earthquake.

Step 2: Express the new intensity in terms of \(S\)

We are looking for the magnitude of an earthquake that is \(10\) times more intense than a standard earthquake. This means:

$$ I = 10S $$

Step 3: Substitute the expression into the formula

Substitute \(I = 10S\) into the original magnitude equation:

$$ M = \log \frac{10S}{S} $$

Answer:

\( M = \log \frac{10S}{S} \)