Why Is a Magnitude 8 Earthquake Not Just a Bit Bigger Than a 7?

Play with the squares before reading on.

• a.Set the earthquakes to 7.0 and 8.0. About how many 7s fit into an 8?
• b.Now try 5.0 and 6.0, then 8.0 and 9.0. What do you notice?
• c.Set 6.0 and 8.0. How many times the energy is two steps? How does that number come from one step’s?
• d.Set Gorkha, 7.8, and Bihar–Nepal, 8.0. How many times the energy?
• e.Find the gap that roughly doubles the energy.

So a magnitude is not a measure like kilograms. It is a scale where adding on the scale means multiplying in the world: adding 1 multiplies the energy by 32, adding 0.2 by 2. Every scale like that is built on one idea, and it has a name.

In 1935 a young seismologist in California, Charles Richter, had a problem. He was measuring how far the pen of a seismograph swung when an earthquake’s waves reached it, to rank California’s earthquakes by size. The swings spread over an enormous range: one earthquake’s could be thousands of times another’s, and the biggest threw the pen beyond what the instrument could record. Numbers that spread so wide are awkward to compare, or to fit in one table.

His colleague Beno Gutenberg suggested a way out: don’t write the swing, write how many times you multiply by 10 to get it. Richter set magnitude 0 as a swing of 1 micrometer (a thousandth of a millimeter) on his standard seismograph, 100 km from the earthquake. Then:

Swing of the pen, 100 km away	As a power of 10 (micrometers)	Magnitude
1 micrometer	10⁰	0
1 millimeter = 1,000 micrometers	10³	3
1 centimeter = 10,000 micrometers	10⁴	4
10 centimeters	10⁵	5
1 meter	10⁶	6

The magnitude is the power of 10: a swing ten times bigger is one more magnitude. (A real pen can’t swing a meter, and seismographs at other distances get corrections; but the idea is this table.)

Logarithms

The power you must raise 10 to, to get a number, is that number’s logarithm (to base 10), written $\log$:

$$\log N = x \quad\text{means}\quad 10^x = N.$$

So $\log 1{,}000 = 3$, because $10^3 = 1{,}000$. Logarithms are the indices of Grades 9 and 10 read backwards: $10^3$ asks what 10 to the power 3 is; $\log 1{,}000$ asks to what power 10 must be raised to make 1,000. The index laws give a few at once:

• $\log 1 = 0$, since $10^0 = 1$. An earthquake of magnitude 0 is real: it is just small.
• $\log 0.01 = -2$, since $10^{-2} = 0.01$. Numbers smaller than 1 have negative logarithms, and earthquakes of negative magnitude are real: instruments close by record them, and a few very shallow ones of about −0.7 have even been felt.
• $\log 0$ doesn’t exist: no power of 10 is 0. So there is no magnitude for “no earthquake at all”, only smaller and smaller ones.

Try these with the line.

• a.Start at 1 and press × 2 again and again. Are the jumps the same length?
• b.Start again and press × 10. Where does 1 → 10 land, and how long is it compared with the × 2 jumps?
• c.Drag the point to 2, then 4, then 8 (watch the number). What logarithms do they have? What do you notice?

Multiplying becomes adding

That last answer is the most useful fact about logarithms. From Grade 8 you know that $10^a \times 10^b = 10^{a + b}$: to multiply powers of 10, add the powers. Logarithms are the powers, so

$$\log (A \times B) = \log A + \log B.$$

The logarithm of a product is the sum of the logarithms. Multiplying in the world becomes adding on the scale. That is exactly what the squares showed: adding 1 to the magnitude multiplies the swing by 10, and two steps of 32 make 32 × 32.

Working out log 2 with no calculator

A puzzle: why is $\log 2$ about 0.301? Look at $2^{10} = 1{,}024$, very nearly $1{,}000 = 10^3$. So ten × 2 jumps on the log line are about as long as three × 10 jumps, and one × 2 jump is about $\tfrac{3}{10}$ of a × 10 jump. (1,024 is a little over 1,000, so $\log 2$ is a little over 0.3: it is 0.30103.)

From that one number, the rest follow by the rules above:

• $\log 4 = \log (2 \times 2) = 0.301 + 0.301 = 0.602$.
• $\log 5 = \log (10 \div 2) = 1 - 0.301 = 0.699$.
• $\log 32 = \log (2^5) = 5 \times 0.301 = 1.505$.

And there is the earthquake’s 32. Its logarithm is about 1.5, so 32 is about $10^{1.5}$, which is $\sqrt{1{,}000} = 31.6$. One magnitude step multiplies the energy by $10^{1.5}$; two steps by $10^{3} = 1{,}000$, as the squares showed.

The seismograph’s swing goes up 10 times a step. Why does the energy go up 32 times? Because an earthquake’s energy grows faster than the height of its waves: as the swing to the power 1.5, so ten times the swing means $10^{1.5} = 31.6$ times the energy. Beno Gutenberg and Charles Richter put it as a formula, for the energy E in joules:

$$\log E \approx 4.8 + 1.5M.$$

It is a rough rule (the USGS gives a slightly different one, $\log E = 5.24 + 1.44M$; for the earthquakes on this page the two agree to within about a quarter), but its 1.5 is the 32. Two earthquakes whose magnitudes differ by d therefore differ in energy by $10^{1.5d}$, and every comparison on this page is that one formula.

So far we went from a magnitude gap to an energy ratio. Turn it round: how big a gap gives a million times the energy? Now the power is the unknown:

$$10^{1.5d} = 1{,}000{,}000 = 10^6, \quad\text{so}\quad 1.5d = 6, \quad d = 4.$$

That was easy only because a million is a whole power of 10. In Grade 10 you solved exponential equations like $2^x = 32$ the same way, by writing both sides as powers of the same number. Logarithms solve the ones where that can’t be done.

The same move, taking logarithms, solves any equation where the unknown is a power. It needs one more rule, from the summary: $\log (a^t) = t \log a$. It holds for any power t, not only whole ones: $a = 10^{\log a}$, so by the index law $a^t = (10^{\log a})^t = 10^{t \log a}$, and the power of 10 that makes $a^t$ is $t \log a$.

The logarithm as a function

Every number x has a power $10^x$, and every positive number has a logarithm, so both are functions, and each undoes the other: $\log (10^x) = x$ and $10^{\log x} = x$. Functions that undo each other are inverse functions (Optional Mathematics, Grade 9), and the graph of an inverse is the reflection of the graph in the line y = x (Figure 1.1): the point (1, 10) on $y = 10^x$ becomes (10, 1) on $y = \log x$.

$10^x$ races upward: every 1 to the right multiplies the height by 10. $\log x$ crawls: to climb from 1 to 2 it must go from 10 to 100, and to reach 3, all the way to 1,000. That crawl is why a log scale fits the huge and the tiny on one page. And $\log x$ has no graph for x ≤ 0.

Base 10 isn’t the only one: $\log_2 N$ counts doublings ($\log_2 8 = 3$), and mathematicians’ favorite is base e = 2.718…, the natural logarithm $\ln$ (the topic on e, in Where to go next).

Earthquakes are not the only things that spread over such a range. Put five earthquakes that shook Nepal between 1934 and 2023, the Assam–Tibet earthquake of 1950 and Valdivia on one ordinary chart, and see what happens.

On an ordinary scale Gorkha, which killed nearly 9,000 people, is a dot squeezed against 0. On a log scale all seven can be compared, and the dots go up evenly with the magnitude, because $\log E = 4.8 + 1.5M$: the magnitude is a position on a log scale.

Scientists reach for a logarithm whenever what they measure spreads over many powers of 10 and what matters is “how many times as big”. Three you will meet:

• pH, for acids. The Danish chemist Søren Sørensen introduced it in 1909: pH is minus the logarithm of the concentration of hydrogen ions (in moles per liter), $\text{pH} = -\log [\text{H}^+]$. The minus sign makes it backwards: a lower pH is more acid. Pure water at 25 ∘°C has pH 7, lemon juice about 2.2. A gap of 4.8 means $10^{4.8} \approx 63{,}000$ times the hydrogen ions.
• Decibels, for loud sounds. A sound’s level is $10 \log (I / I_0)$ decibels, where I is the sound’s intensity and I₀ a standard, $10^{-12}$ watts per square meter. Every 10 decibels is 10 times the intensity, so 90 dB is 1,000 times 60 dB, and every 3 dB about doubles it ($10^{0.3} \approx 2$).
• Star magnitudes. Ancient astronomers sorted the stars into six magnitudes, the brightest first. In 1856 Norman Pogson made it exact: a first-magnitude star is 100 times as bright as a sixth, five steps, so each step is the fifth root of 100, about 2.512 times (2.512 multiplied by itself five times makes 100). Backwards again: brighter stars have smaller magnitudes, and the brightest go negative. Sirius, the brightest star at night, is −1.46; the faintest stars you can see on a dark night are about +6.5, about 1,500 times fainter.

The magnitude scale has another surprise, found by counting. The USGS catalog lists every earthquake of magnitude 6 or more in the world. Over the 40 years from 1980 to 2019:

Magnitude	Earthquakes in 40 years	A year, on average
6 to 6.9	5,298	132
7 to 7.9	533	13
8 or more	34	under 1

From each band to the next, about 10 times fewer. Gutenberg and Richter found this in 1944 for California, then worldwide: the number of earthquakes of at least a magnitude falls 10 times with each step up. It is called the Gutenberg–Richter law, and its own form is a logarithm: $\log N = a - bM$, where N is how many earthquakes of magnitude M or more, and b is usually close to 1. Nepal’s aftershocks of 2015 followed it. By 24 May 2016 there had been 459 aftershocks of magnitude 4 or more, 51 of 5 or more, and 5 above 6: 459, 51, 5, about ten times fewer each step.

Now put the two rules together. Each step up there are 10 times fewer earthquakes, but each one releases 32 times the energy. So the bigger earthquakes, though rare, release more energy between them: about 32 ÷ 10 = 3 times more each band. The few giants carry most of the energy.

A scale that runs out

Richter’s scale works by the swing of a pen, and the pen can’t swing for ever. For big earthquakes, waves that are short beside a fault hundreds of kilometers long stop growing, and the old scales saturate: Richter’s own stops at about 7, the surface-wave scale used in the 1970s at about 8 to 8.5. On that scale the Valdivia earthquake measured only about 8.2, though it was far bigger than any other 8.2.

In 1979 Thomas Hanks and Hiroo Kanamori defined the moment magnitude, $M_w$, which the USGS now uses for big earthquakes. It doesn’t measure a swing at all. It measures what the fault did, its seismic moment:

$$M_0 = \text{rigidity of the rock} \times \text{area that slipped} \times \text{distance it slipped}, \qquad M_w = \tfrac{2}{3}\left(\log M_0 - 9.1\right),$$

with M₀ in newton meters. A logarithm again, chosen to agree with Richter’s scale where both work. So a bigger fault and more slip mean a bigger magnitude, without a limit set by any instrument.

The scale has no top, but the Earth does: the USGS works out that a magnitude 10 would need a fault about 14,000 km long and 100 km wide to slip by 30 meters. South America’s whole subduction zone, 6,400 km, slipping 40 meters end to end, would make a 9.86.

One magnitude, many shakings

The magnitude is one number for the whole earthquake. How hard it shakes you, its intensity, depends on how far away you are and on the ground under you. And the magnitude is measured, not given: for Gorkha the USGS first said 7.5, then 7.8; the Global Centroid Moment Tensor project said 7.9. A difference of 0.3 sounds small, and by now you know it isn’t: $10^{0.45} \approx 2.8$ times the energy.

• Seismology. Earthquake reports give a moment magnitude, and seismologists fit the Gutenberg–Richter law’s a and b to each region’s record, which differ from region to region. Nepal’s National Seismological Centre counts and locates the country’s earthquakes.
• Chemistry, pH (above). The first electronic pH meter was made in 1934 by Arnold Beckman at the California Institute of Technology, for a lemon grower who wanted to test lemons quickly.
• Sound, decibels (above).
• Astronomy, star magnitudes (above): the Sun is −26.8, the faintest things the Hubble Space Telescope has photographed about +31.5.
• Dating the past, with half-lives: Willard Libby’s radiocarbon dating (the bone above) won him the Nobel Prize in Chemistry in 1960.

• How Did a Book of Numbers Make Multiplying Easy? Napier’s logarithms of 1614, the tables that turned every multiplication into an addition, and the slide rule.
• What Is e, and Why Does Growth Keep Finding It? Interest added more and more often, and the natural logarithm.
• Why Does a Scale Have Twelve Notes? Music’s octaves, a log scale your ears use.
• What Is the Biggest Number Anyone Has Used? When even logarithms aren’t enough.
• How Much Rice Is on the Chessboard? (Grades 6–8) Doubling 64 times, the growth that a logarithm undoes.
• Computing, Going Further: How Many Moves to Move 64 Disks?, another doubling that runs away, and how a computer counts it; and How Often Does Nepal Shake?, Nepal’s earthquake record as data. (Coming soon.)
• Science, Going Further: Can We Tell When Disaster Will Strike? and Can You Design an Earthquake-Safe House?; and Earthquakes and Plates, every earthquake of magnitude 6 or more since 1900, on Our World.
• Grade 10, Indices (exponential equations) and Growth and Depreciation.