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By what factor does the intensity increase for each whole-number increase in the Richter scale? Use your understanding of logarithms and the inverse relationship between logarithms and exponents to explain your answer.
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The Richter magnitude M is defined logarithmically (base 10) from intensity I, for example
M = log10(I/I0),
so by the inverse relationship between logs and exponents,
I = I0 · 10^M.
If the magnitude increases by 1 (M → M+1), the new intensity is
I' = I0 · 10^(M+1) = (I0 · 10^M) · 10 = I · 10.
So each whole-number increase on the Richter scale corresponds to a 10-fold increase in measured ground-motion intensity.
(As a related note: the seismic energy released scales differently — roughly E ∝ 10^(1.5M) — so the energy released rises by about 10^1.5 ≈ 31.6 times for each unit increase in magnitude.)
M = log10(I/I0),
so by the inverse relationship between logs and exponents,
I = I0 · 10^M.
If the magnitude increases by 1 (M → M+1), the new intensity is
I' = I0 · 10^(M+1) = (I0 · 10^M) · 10 = I · 10.
So each whole-number increase on the Richter scale corresponds to a 10-fold increase in measured ground-motion intensity.
(As a related note: the seismic energy released scales differently — roughly E ∝ 10^(1.5M) — so the energy released rises by about 10^1.5 ≈ 31.6 times for each unit increase in magnitude.)
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