Question

Consider the equation =2/3log10()−10.7 relating the moment magnitude of an earthquake and the energy (in ergs) released by it. If increases by 6, by what factor does the energy increase?

Answer 1

The energy, 'E', increases by a factor of 1000,000,000

Explanation:

The given equation relating moment magnitude of an earthquake and the energy (in ergs) released by it obtained from a similar question online is presented as follows;
`M_W = (2)/(3) * log_(10) (E) - 10.7`

Where;

`M_W` = The moment magnitude

E = The energy

If
`M_W` is increased by 6, we have;

Δ
`M_W` = 6

`\Delta M_W = M_(W2) - M_(W1) = 6`

From which we get;

`\Delta M_W = M_(W2) - M_(W1) = 6 = (2)/(3) * log_(10) (E_2) - 10.7 - \left((2)/(3) * log_(10) (E_1) - 10.7 \right)`

`6 = (2)/(3) * log_(10) (E_1) - 10.7 - \left((2)/(3) * log_(10) (E_2) - 10.7 \right) = \left((2)/(3) \right ) * \left ( log_(10) (E_2) - log_(10) (E_1) \right)`

`6 = \left((2)/(3) \right ) * \left ( log_(10) (E_2) - log_(10) (E_1) \right) = \left((2)/(3) \right ) * log_(10) \left( (E_2)/(E_1) \right)`

`6 = \left((2)/(3) \right ) * log_(10) \left( (E_2)/(E_1) \right)`

`log_(10) \left( (E_2)/(E_1) \right) = 6 * \left((3)/(2) \right ) = 9`

Therefore;

`(E_2)/(E_1) = 10^9`

E₂ = E₁ × 10⁹ = 1000,000,000 × E₁

Therefore, when the moment magnitude,
`M_W`, increases by 6, the energy increases by a factor of 1000,000,000