Comparison between two earthquakes

Statement

Two earthquakes have Richter magnitudes $M_1 = 5{,}5$ and $M_2 = 7{,}5$. How many times greater is the amplitude of the waves of the second compared with the first? And the energy?

Solution

Amplitude. From $M = \log_{10}(A/A_0)$ it follows that $A = A_0\cdot 10^{M}$, hence $\frac{A_2}{A_1} = 10^{\,M_2 - M_1} = 10^{\,7{,}5 - 5{,}5} = 10^{2} = 100.$ The amplitude of the second is $\boxed{100}$ times that of the first.

Energy. The energy grows as $10^{1{,}5 M}$, therefore $\frac{E_2}{E_1} = 10^{\,1{,}5(M_2 - M_1)} = 10^{\,1{,}5\cdot 2} = 10^{3} = 1000.$ The second earthquake releases $\boxed{1000}$ times the energy of the first.

Links

Topics: Logarithmic function
Concepts: Logarithm · Logarithmic scale · Richter scale
Functions: Logarithmic function
Skills: Calculating · Modelling
Exercise type: Modelling problem