To determine the magnitude [tex]\( M \)[/tex] of an earthquake that is 1,000 times more intense than a standard earthquake, follow these steps:
1. Identify the given values:
- The intensity of the earthquake, [tex]\( I \)[/tex], is 1,000 times more intense than a standard earthquake.
- The intensity of a standard earthquake, [tex]\( S \)[/tex], is the reference value.
Therefore, [tex]\( I = 1000 \times S \)[/tex].
2. Use the formula for the magnitude [tex]\( M \)[/tex]:
The magnitude [tex]\( M \)[/tex] is given by:
[tex]\[
M = \log_{10} \left( \frac{I}{S} \right)
\][/tex]
3. Substitute the given values into the formula:
Since [tex]\( I = 1000 \times S \)[/tex], we can substitute this into the formula:
[tex]\[
M = \log_{10} \left( \frac{1000 \times S}{S} \right)
\][/tex]
4. Simplify the fraction:
[tex]\[
M = \log_{10} \left( 1000 \right)
\][/tex]
5. Calculate the logarithm:
[tex]\[
M = \log_{10} 1000
\][/tex]
We know that [tex]\( 10^3 = 1000 \)[/tex], so:
[tex]\[
\log_{10} 1000 = 3
\][/tex]
6. Round the answer to the nearest tenth:
Since 3 is already an integer, it can be represented as 3.0 when rounded to the nearest tenth.
Therefore, the magnitude of the earthquake is:
[tex]\[
M = 3.0
\][/tex]
So, the correct answer is:
[tex]\[ 3 \][/tex]
