To find the Richter scale rating [tex]\( R \)[/tex] for an amplitude of [tex]\( 100,000,000 I_0 \)[/tex], we will use the formula for the magnitude on the Richter scale:
[tex]\[ R = \log_{10} \left( \frac{I}{I_0} \right) \][/tex]
Given that:
[tex]\[ I = 100,000,000 I_0 \][/tex]
we substitute [tex]\( I \)[/tex] into the formula for [tex]\( R \)[/tex]:
[tex]\[ R = \log_{10} \left( \frac{100,000,000 I_0}{I_0} \right) \][/tex]
Since [tex]\( I_0 \)[/tex] appears in both the numerator and the denominator, it cancels out:
[tex]\[ R = \log_{10} \left( 100,000,000 \right) \][/tex]
Next, we recognize that [tex]\( 100,000,000 \)[/tex] can be written as [tex]\( 10^8 \)[/tex]:
[tex]\[ R = \log_{10} \left( 10^8 \right) \][/tex]
The logarithmic property [tex]\( \log_{10}(10^a) = a \)[/tex] tells us that:
[tex]\[ R = 8 \][/tex]
Therefore, the magnitude [tex]\( R \)[/tex] on the Richter scale for an amplitude of [tex]\( 100,000,000 I_0 \)[/tex] is:
[tex]\[ R = 8 \][/tex]
