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Logarithmic measures

Measurement · level 8, lesson 6 of 6

A question to start

A busy street is about 80 decibels. A quiet conversation is about 60. The street is only 20 decibels louder. So is it a third louder? A third more sound energy?

No. The street delivers about 100 times as much sound energy to your ear each second. And a magnitude 7 earthquake is not "a bit stronger" than a magnitude 5: it releases about 1000 times as much energy. Lemon juice, at pH 2, is not "a bit more acidic" than milk, at about pH 6.5: it has tens of thousands of times as many acid particles per litre.

Decibels, earthquake magnitudes and pH are all logarithmic measures. On each scale, a step of 1 does not mean "add 1". It means "multiply by a fixed factor". This lesson shows how they work, why they are used, and how to move between the scale and the real quantity.

The idea

When numbers cover too many powers of ten

Some quantities come in an enormous range. The quietest sound a young person can hear carries about \(10^{-12}\) watts per square metre to the ear; a sound loud enough to hurt carries about 1 W/m². That is a range of a million million (\(10^{12}\)). On an ordinary number line, every quiet sound would be squashed into an invisible speck next to zero.

The answer is to measure how many powers of ten a quantity is, rather than the quantity itself. That is exactly what a logarithm does (Logarithms):

\[\log_{10} x = y \quad \text{means} \quad 10^y = x.\]

So \(\log_{10} 1000 = 3\), \(\log_{10} 10^{-12} = -12\), and \(\log_{10} 2 \approx 0.301\). The key law is

\[\log_{10}(a \times b) = \log_{10} a + \log_{10} b.\]

Multiplying the quantity adds to the logarithm. That is what turns "times 10" into "plus 1". It is the ×10 ruler from Fermi estimates, now with a formula.

Decibels: sound level

The sound intensity \(I\) is the power of sound passing through each square metre, in W/m². The sound level in decibels (dB) compares it with a reference intensity \(I_0 = 10^{-12}\) W/m², about the threshold of hearing:

\[L = 10 \log_{10}\!\left(\frac{I}{I_0}\right) \text{ dB}.\]

Why the 10? The original unit was the bel, one power of ten. A bel was too coarse, so it is split into ten decibels.

0 dB204060 80100120 10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶ 10⁻⁴10⁻²1 W/m² hearingwhispertalking streetpain

Read the scale:

  • +10 dB means × 10 intensity. (Add 1 to the logarithm, multiply by 10.)
  • +20 dB means × 100. That is the street versus the conversation.
  • +3 dB means about × 2, because \(10 \log_{10} 2 \approx 3.01\).

Our ears respond roughly to ratios, not differences, so a log scale matches how we hear better than a linear one would: by a common rule of thumb, each extra 10 dB sounds roughly "twice as loud", even though it carries 10 times the energy.

Adding sources. Two identical 60 dB speakers do not make 120 dB. Their intensities add: \(2I\). So the level goes up by \(10 \log_{10} 2 \approx 3\) dB, to about 63 dB. To add levels, change back to intensities, add, and take the logarithm again.

Earthquake magnitudes

In 1935 the seismologist Charles Richter, in California, made a scale for local earthquakes. Its magnitude is the logarithm of the largest wave amplitude (how far the ground swings) recorded on a standard seismometer, corrected for the distance to the quake. So each step of 1 is 10 times the amplitude.

The energy released grows faster than the amplitude. Seismologists found that energy goes up by a factor of about \(10^{1.5} \approx 31.6\) for each step of magnitude. So:

  • 1 step: about 32 times the energy;
  • 2 steps: \(10^{1.5 \times 2} = 10^3 = 1000\) times;
  • 0.2 steps: \(10^{0.3} \approx 2\) times. A magnitude 6.2 releases about twice the energy of a 6.0.

The Richter scale stops working well for very large earthquakes (its readings stop growing). Today scientists use the moment magnitude \(M_w\) (introduced in 1979 by Thomas Hanks and Hiroo Kanamori). It starts from the seismic moment \(M_0\), a physical size of the quake in newton-metres: roughly, the stiffness of the rock times the area of the fault that slipped times how far it slipped. Then

\[M_w = \frac23\left(\log_{10} M_0 - 9.1\right).\]

The constants were chosen so that \(M_w\) agrees with the older scales where they overlap. Each step of 1 in \(M_w\) is a factor of \(10^{1.5}\) in \(M_0\) (solve for \(M_0\) to see why). The largest earthquake ever recorded, in Chile in 1960, had \(M_w\) about 9.5.

Earth is not the only planet with quakes. NASA's InSight lander placed a seismometer on Mars and recorded over a thousand marsquakes between 2018 and 2022. The largest, in May 2022, was about magnitude 4.7. Because the scale is logarithmic, that one number tells us at once how it compares with Earth's great earthquakes.

pH: acidity

How acidic a liquid is depends on the concentration of hydrogen ions, written \([\text{H}^+]\), measured in moles per litre (mol/L; the mole is an SI base unit, see the last lesson). These concentrations range from about 1 mol/L down to \(10^{-14}\) mol/L. So chemists use

\[\text{pH} = -\log_{10}[\text{H}^+].\]

The minus sign makes the usual pH values positive.

  • Pure water at 25 °C: \([\text{H}^+] = 10^{-7}\) mol/L, so pH 7: neutral.
  • Below 7 is acidic (more H⁺): lemon juice is about pH 2.
  • Above 7 is basic (alkaline): household bleach is about pH 12 to 13.

Because of the minus sign, lower pH means more acid, and each step down of 1 means 10 times the concentration of hydrogen ions.

The pattern

All three scales have the same shape:

\[\text{measure} = (\text{constant}) \times \log_{10}\!\left(\frac{\text{quantity}}{\text{reference}}\right).\]
Scale Constant One step means
decibels 10 +10 dB is × 10 intensity
moment magnitude \(\tfrac23\) +1 is × \(10^{1.5} \approx 32\) in moment (and energy)
pH \(-1\) −1 is × 10 hydrogen-ion concentration

To go from the scale back to the quantity, undo the logarithm with a power of ten:

\[I = I_0 \times 10^{L/10}, \qquad [\text{H}^+] = 10^{-\text{pH}}, \qquad M_0 = 10^{1.5 M_w + 9.1}.\]

Comparing two readings never needs the reference. The difference of two logarithms is the logarithm of the ratio: \(L_2 - L_1 = 10 \log_{10}(I_2 / I_1)\). That is why "20 dB louder" always means "100 times the intensity", whatever the starting level.

Worked examples

1. Sound level from intensity. A street has intensity \(10^{-4}\) W/m². Find the level.

\(L = 10 \log_{10}\!\left(\dfrac{10^{-4}}{10^{-12}}\right) = 10 \log_{10}(10^8) = 10 \times 8 = 80\) dB.

2. Intensity from sound level. A concert is 110 dB. Find its intensity, and compare it with a 60 dB conversation.

  1. \(I = 10^{-12} \times 10^{110/10} = 10^{-12} \times 10^{11} = 10^{-1} = 0.1\) W/m².
  2. Ratio: \(10^{(110 - 60)/10} = 10^5\). The concert carries 100 000 times the intensity of the conversation.

3. Two magnitudes. How much more energy does a magnitude 7.0 earthquake release than a magnitude 5.0?

Difference 2.0 steps: \(10^{1.5 \times 2.0} = 10^{3} = 1000\) times.

4. Moment magnitude. A quake has seismic moment \(M_0 = 4 \times 10^{22}\) N·m. Find \(M_w\).

  1. \(\log_{10}(4 \times 10^{22}) = \log_{10} 4 + 22 \approx 0.602 + 22 = 22.602\).
  2. \(M_w = \tfrac23 (22.602 - 9.1) = \tfrac23 \times 13.502 \approx 9.0\).

5. pH. Lemon juice has \([\text{H}^+] \approx 3.2 \times 10^{-3}\) mol/L. Find its pH, and compare with pure water.

  1. \(\text{pH} = -\log_{10}(3.2 \times 10^{-3}) = -(\log_{10} 3.2 - 3) \approx -(0.505 - 3) \approx 2.5\).
  2. Water has pH 7: the difference is 4.5, so lemon juice has \(10^{4.5} \approx 32\,000\) times the concentration of H⁺.

Try it

  1. Find the sound level of an intensity of \(10^{-9}\) W/m².
  2. What intensity gives 100 dB?
  3. A lawnmower is 90 dB and a library is 40 dB. How many times greater is the lawnmower's intensity?
  4. One machine makes 70 dB. What level do (a) two, and (b) ten identical machines make together?
  5. Compare a magnitude 6.0 and a magnitude 4.0 earthquake: by what factor do the amplitudes differ, and the energies?
  6. Find the pH when \([\text{H}^+] = 4 \times 10^{-4}\) mol/L. Is it acidic or basic?
  7. Sea water has pH about 8.1. Find \([\text{H}^+]\).
  8. The largest marsquake InSight recorded was about magnitude 4.7, and the 1960 Chile earthquake about 9.5. Roughly how many times more energy did the Chile earthquake release?
Answers
  1. \(10 \log_{10}(10^{-9} / 10^{-12}) = 10 \log_{10}(10^3) = 30\) dB.
  2. \(I = 10^{-12} \times 10^{10} = 10^{-2} = 0.01\) W/m².
  3. Difference \(50\) dB, so \(10^{50/10} = 10^5 = 100\,000\) times.
  4. (a) Intensities double: \(70 + 10\log_{10} 2 \approx 73\) dB. (b) Intensities ×10: \(70 + 10 = 80\) dB.
  5. Amplitude: \(10^{2} = 100\) times. Energy: \(10^{1.5 \times 2} = 1000\) times.
  6. \(-\log_{10}(4 \times 10^{-4}) = -(0.602 - 4) \approx 3.4\). Below 7, so acidic.
  7. \([\text{H}^+] = 10^{-8.1} \approx 7.9 \times 10^{-9}\) mol/L.
  8. Difference \(9.5 - 4.7 = 4.8\) steps: \(10^{1.5 \times 4.8} = 10^{7.2} \approx 1.6 \times 10^7\), about 16 million times as much energy.

Common mistake

Reading a log scale as if it were linear. "80 dB is a third more than 60 dB" and "magnitude 8 is twice magnitude 4" are both wrong. On a log scale, a difference in readings stands for a ratio in the real quantity. Always turn the difference into a power of ten.

Adding decibels directly. Two 60 dB sources make about 63 dB, not 120 dB. Add intensities, then take the logarithm.

Where it leads