In case of earthquakes, an increase of magnitude 1 on Richter Scale implies

Concept: The Richter magnitude of an earthquake is a base-10 logarithmic measure of the amplitude of the seismic waves it generates, written as

M = log10(A/A0)

Here A is the maximum trace amplitude recorded by a standard seismograph and the denominator is a fixed reference amplitude. Because the magnitude is a logarithm of amplitude, equal additive steps on the scale correspond to equal multiplicative factors in amplitude, never to equal additive increases.

Application: compare two earthquakes whose magnitudes differ by exactly one unit.

1. Write each earthquake's magnitude from its own recorded amplitude:

   M1 = log10(A1/A0) and M2 = log10(A2/A0)

2. Subtract the smaller magnitude from the larger one; the fixed reference amplitude cancels:

   M2 − M1 = log10(A2/A1)

3. Set that difference equal to the one whole unit given in the question:

   log10(A2/A1) = 1

4. Raise 10 to the power of each side to undo the logarithm:

   A2/A1 = 101 = 10

5. The larger earthquake therefore traces an amplitude ten times that of the smaller one, so a rise of one Richter unit means a ten-fold increase in the amplitude of the seismic waves.

Cross-check and contrast:

• Released energy does not scale by the same factor. The standard energy-magnitude relation is log10E = 4.8 + 1.5M ⇒ E2/E1 = 101.5 ≈ 31.6 so a one-unit magnitude step multiplies the released energy by about 31.6, not by 10.

• Doubling the amplitude is a far smaller change on the scale, because log102 ≈ 0.30 so it moves the magnitude by only about a third of one unit.

• Doubling the released energy is a smaller step still: the same energy relation gives ΔM = (log102) / 1.5 ≈ 0.20 so twice the energy is only about a fifth of one magnitude unit.

• Substituting the result back into the definition confirms it: log1010 = 1 which is exactly the one-unit step described in the question.