To solve for [tex]\( I \)[/tex] in terms of [tex]\( I_0 \)[/tex], we can follow these steps:
1. Understand the Richter scale formula:
[tex]\[
\text{Magnitude} = \log_{10} \left(\frac{I}{I_0}\right)
\][/tex]
Given, the magnitude is 6.8.
2. Write down the given magnitude equation:
[tex]\[
6.8 = \log_{10} \left(\frac{I}{I_0}\right)
\][/tex]
3. Solve for the ratio [tex]\(\frac{I}{I_0}\)[/tex]:
[tex]\[
\frac{I}{I_0} = 10^{6.8}
\][/tex]
4. Calculate [tex]\( 10^{6.8} \)[/tex]:
[tex]\[
10^{6.8} \approx 6309573.44480193
\][/tex]
5. Express [tex]\( I \)[/tex] in terms of [tex]\( I_0 \)[/tex]:
[tex]\[
I = 6309573.44480193 \times I_0
\][/tex]
6. Round the value to the nearest whole number:
[tex]\[
I \approx 6309573 \times I_0
\][/tex]
Therefore, the amplitude [tex]\( I \)[/tex] can be expressed in terms of [tex]\( I_0 \)[/tex] as:
[tex]\[
I \approx 6309573 \, I_0
\][/tex]
So, the final answer is:
[tex]\[
I \approx 6309573 \, I_0
\][/tex]
