Question

The Richter scale magnitude, R
, of an earthquake of intensity I
is defined as R=log(II0),
where I0
is a small threshold intensity. Find the magnitude of an earthquake with intensity 55,000,000I0
.

Answer 1

To find the magnitude \( R \) of an earthquake with intensity \( I = 55,000,000 I_0 \), we will use the formula:

\[
R = \log\left(\frac{I}{I_0}\right)
\]

Substituting the value of \( I \):

\[
R = \log\left(\frac{55,000,000 I_0}{I_0}\right)
\]

The \( I_0 \) terms cancel out:

\[
R = \log(55,000,000)
\]

Now we need to calculate \( \log(55,000,000) \). We can express \( 55,000,000 \) as \( 5.5 \times 10^7 \):

\[
R = \log(5.5 \times 10^7)
\]

Using the logarithmic property \( \log(ab) = \log(a) + \log(b) \):

\[
R = \log(5.5) + \log(10^7)
\]

Since \( \log(10^7) = 7 \):

\[
R = \log(5.5) + 7
\]

Now we need the value of \( \log(5.5) \). Approximating this using a scientific calculator or logarithm table, we find:

\[
\log(5.5) \approx 0.740
\]

Putting it all together:

\[
R \approx 0.740 + 7 = 7.740
\]

Thus, the magnitude of the earthquake is approximately:

\[
\boxed{7.74}
\]