A. Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Logarithmic Scales: pH, Decibels and Earthquake Magnitude

A2.4 A3.4 A1.2About 75 minutes

Learning goals

I can explain why scientists use logarithmic scales for quantities that vary over huge ranges.
I can calculate and interpret pH, sound level in decibels and earthquake magnitude.
I can compare two quantities on a log scale using the rule "a difference of d on the scale is a ratio of 10ᵈ".
I can solve problems that require switching between logarithmic and exponential form.

Words to know

logarithmic scale
A scale on which each step of 1 multiplies the underlying quantity by the same factor (usually 10).

pH
pH = −log [H⁺], where [H⁺] is the hydrogen ion concentration in moles per litre. Lower pH means more acidic.

decibel (dB)
A unit of sound level: β = 10 log (I/I₀), where I is the sound intensity in W/m² and I₀ = 10⁻¹² W/m² is about the quietest sound a person can hear.

magnitude (earthquake)
A logarithmic measure of an earthquake's size. On the Richter scale M = log (A/A₀), where A is the amplitude of the seismograph trace and A₀ is a standard small amplitude.

moment magnitude
The scale seismologists usually report today. It is calculated from the energy-related size of the fault slip, but it was designed to give values close to the Richter scale.

Numbers too big (and too small) for an ordinary scale

The loudest sound you can stand is about a trillion times more intense than the quietest one you can hear. The hydrogen ion concentration in lemon juice is about 100 000 times that of pure water. A great earthquake shakes the ground thousands of times harder than a small one.

On an ordinary scale, everything interesting would be squashed into one corner. A logarithmic scale solves the problem: it records the exponent. Each step of 1 on the scale means multiplying by 10. That is why logarithms appear all over science, and why this lesson ties together everything you have learned in the unit.

The one rule to remember

On a base-10 log scale, a difference of d in the scale value means a ratio of 10ᵈ in the underlying quantity. (For decibels, which use 10 log, the ratio is 10^(d/10).) This follows from the quotient law: log A − log B = log (A/B).

pH: acidity

Chemists measure acidity with pH = −log [H⁺], where [H⁺] is the concentration of hydrogen ions in mol/L. The negative sign makes most pH values positive, because concentrations are usually tiny.

Pure water at 25 °C has [H⁺] = 10⁻⁷ mol/L, so pH = 7 (neutral).
pH below 7 is acidic; pH above 7 is basic (alkaline).
A drop of 1 in pH means 10 times more hydrogen ions.

To go backward: [H⁺] = 10^(−pH).

The pH scale with approximate values for some familiar substances. Lemon juice (about pH 2) has about 10⁵ = 100 000 times the hydrogen ion concentration of pure water (pH 7).

Worked example

Example 1: Finding pH and concentration

(a) A sample of orange juice has [H⁺] = 3.2 × 10⁻⁴ mol/L. Find its pH. (b) A soil sample has pH 6.4. Find [H⁺].

(a) pH = −log (3.2 × 10⁻⁴) ≈ −(−3.495) = 3.49.
(b) [H⁺] = 10^(−6.4) ≈ 3.98 × 10⁻⁷ mol/L.
Reasonable? (a) is between 10⁻⁴ and 10⁻³, so pH between 3 and 4 ✓

(a) pH ≈ 3.49 (b) [H⁺] ≈ 3.98 × 10⁻⁷ mol/L

Worked example

Example 2: Comparing acidity

Normal rain is slightly acidic, with pH about 5.6, because carbon dioxide from the air dissolves in it. Rain with a lower pH is called acid rain. How many times more acidic (in [H⁺]) is acid rain with pH 4.3 than normal rain?

Difference in pH: 5.6 − 4.3 = 1.3.
Ratio of concentrations: 10^1.3 ≈ 20.0.
Algebra behind it: [H⁺]₁ ÷ [H⁺]₂ = 10^(−4.3) ÷ 10^(−5.6) = 10^(1.3).

About 20 times the hydrogen ion concentration. Acid rain harmed many lakes in Ontario's Canadian Shield region in the 1970s and 1980s, which is one reason emissions rules were tightened.

Decibels: loudness

Sound level is β = 10 log (I/I₀) decibels, where I₀ = 10⁻¹² W/m².

The threshold of hearing, I = I₀, gives 10 log 1 = 0 dB.
Every 10 dB increase means 10 times the intensity.
Doubling the intensity adds 10 log 2 ≈ 3 dB.

Our ears respond roughly to the logarithm of intensity, which is why the decibel scale matches how loud things seem better than raw intensity does. Many workplace safety rules, including Ontario's, use an average sound level of 85 dBA (A-weighted decibels) over an 8-hour shift as a limit, because long exposure to louder sound can damage hearing.

Each 10 dB step up the ladder multiplies the sound intensity by 10. From 0 dB to 120 dB the intensity grows by a factor of 10¹², a trillion. The everyday examples are approximate.

Worked example

Example 3: Decibel calculations

(a) A vacuum cleaner has intensity 10⁻⁵ W/m². Find its sound level. (b) How many times more intense is 85 dB than 60 dB? (c) Two identical speakers each produce 70 dB. What is the level when both play?

(a) β = 10 log (10⁻⁵ ÷ 10⁻¹²) = 10 log 10⁷ = 70 dB.
(b) Difference 25 dB. Ratio 10^(25/10) = 10^2.5 ≈ 316.
(c) Doubling the intensity: 70 + 10 log 2 ≈ 70 + 3.0 = 73.0 dB. Not 140 dB!

(a) 70 dB (b) about 316 times (c) about 73 dB

Earthquake magnitude

Charles Richter's scale (1935) defined magnitude as M = log (A/A₀), where A is the amplitude of the seismograph trace (corrected for distance) and A₀ is a standard tiny amplitude.

Each increase of 1 in magnitude means 10 times the amplitude of shaking on the seismograph.
The energy released grows faster: about 10^1.5 ≈ 31.6 times per unit of magnitude.

Today seismologists usually report the moment magnitude for larger earthquakes, which is calculated from the size of the fault slip but was designed to agree closely with Richter's values. The rules "1 unit = 10 times the amplitude" and "1 unit ≈ 32 times the energy" are still good guides.

Worked example

Example 4: Comparing earthquakes

The 2012 earthquake off Haida Gwaii, British Columbia, had moment magnitude 7.8. Compare it with a magnitude 5.8 earthquake: (a) amplitude, (b) energy.

Difference in magnitude: 7.8 − 5.8 = 2.
(a) Amplitude ratio: 10² = 100.
(b) Energy ratio: (10^1.5)² = 10³ = 1000.

The 7.8 earthquake had about 100 times the amplitude and released about 1000 times the energy.

Worked example

Example 5: Working backward

A seismograph records an amplitude 250 times that of a magnitude 4.1 earthquake (at the same distance). What is the magnitude?

Let the magnitudes be M and 4.1. Then M − 4.1 = log (A ÷ A₄.₁) = log 250.
log 250 ≈ 2.398.
M ≈ 4.1 + 2.398 ≈ 6.5.

Magnitude about 6.5

Knowledge that came before the seismographs

The largest known earthquake in Canada's history happened before seismographs existed. First Nations oral histories from the Pacific coast, including accounts passed down in Nuu-chah-nulth communities on the west coast of Vancouver Island, describe a great earthquake that shook the land at night and seawater that flooded coastal villages. These histories remain an important source of knowledge about the event. In Japan, written records describe an "orphan tsunami" that arrived in January 1700 with no earthquake felt there beforehand. Putting the oral histories, the Japanese records and geological evidence together, scientists date this Cascadia earthquake to the evening of 26 January 1700 and estimate its moment magnitude at about 9. Compared with the 7.8 Haida Gwaii earthquake, a difference of about 1.2 means roughly 10^1.2 ≈ 16 times the amplitude and 10^1.8 ≈ 63 times the energy.

Just for fun

The bel, the unit behind the decibel, is named after Alexander Graham Bell, who worked out his idea for the telephone at his family's home in Brantford, Ontario. A decibel is a tenth of a bel, which is why nobody ever says a concert was "11 bels" loud.

Hands-on activity

A red cabbage pH indicator

You need: about 1/4 of a red cabbage, a knife and cutting board, very hot water from a kettle, a large heat-proof bowl, a strainer, 4 clear glasses, and test liquids: water, white vinegar, lemon juice and a solution of 1 teaspoon of baking soda in half a cup of water. Cut on the cutting board with the blade moving away from your fingers, and pour the hot water slowly so it cannot splash. Ask an adult to help if you are unsure. Use only these kitchen liquids; never test cleaning products.

Chop the cabbage, cover it with hot water in the bowl and let it stand for about 15 minutes until the water is deep purple. Let it cool, then strain it.
Pour about 2 cm of cabbage juice into each glass.
Add a spoonful of a different test liquid to each glass. Note the colour.
Using the pH scale diagram, put the liquids in order from most acidic to most basic.
Lemon juice is about pH 2 and the baking soda solution about pH 8. How many times greater is [H⁺] in lemon juice?

What to look for: acids turn the indicator red or pink, the water stays purple-ish, and the baking soda solution turns it blue or green. The order should be lemon juice, vinegar, water, baking soda. The colours give only a rough range, not an exact pH. For step 5, the difference is 6, so the ratio is 10⁶: a million times greater. Pour everything down the drain with plenty of water when you are done.

Your turn

Find the pH of a solution with [H⁺] = 10⁻⁹ mol/L. Is it acidic or basic?

Find the pH of a solution with [H⁺] = 4 × 10⁻⁷ mol/L.

A drink has pH 3.2. Find [H⁺].

How many times more acidic is pH 3 than pH 6?

A sound has intensity 10⁻³ W/m². Find its level in decibels.

How many times more intense is a 100 dB sound than a 70 dB sound?

Compare the amplitudes of earthquakes of magnitude 6.2 and 4.7.

More practice: warm-up

What is the pH of pure water at 25 °C?

Is a solution with pH 4 acidic or basic?

What is the sound level of the threshold of hearing?

How many times more intense is 50 dB than 40 dB?

A magnitude 5 earthquake has how many times the amplitude of a magnitude 4 earthquake?

More practice: on your own

A swimming pool should have pH about 7.4. Find [H⁺].

Find the intensity of a 75 dB sound.

Five identical machines each make 80 dB. What is the level with all five running?

Earthquake A has magnitude 6.4, and earthquake B has 1/300 of A's amplitude. Find B's magnitude.

About how many times more energy does a magnitude 7.0 earthquake release than a 6.4? (Use 10^(1.5d).)

Challenge

Show that if β₁ and β₂ are two sound levels with intensities I₁ and I₂, then β₁ − β₂ = 10 log (I₁/I₂).

Equal volumes of two solutions with pH 2 and pH 4 are mixed. Assuming the hydrogen ions just add up and the volume doubles, find the pH of the mixture. Why is it not 3?

How many identical 60 dB voices would it take to reach 85 dB?

Check your understanding

Quiz

A solution has [H⁺] = 10⁻⁵ mol/L. What is its pH?

How many times greater is [H⁺] at pH 2 than at pH 5?

A sound's intensity is multiplied by 100. How does its level change?

Two identical 80 dB speakers play together. The level is about

An earthquake of magnitude 7 has about how many times the amplitude of one of magnitude 5?

Why are pH, decibels and magnitude logarithmic scales?

For parents and teachers

Big idea (A2.4, A3.4): pH, decibels and earthquake magnitude are logarithmic scales. A difference of d on a base-10 scale means a ratio of 10ᵈ (10^(d/10) for decibels), so a small-looking change can be a huge change in the real quantity.

How to help: Use real numbers from around the house: ask how much more acidic vinegar (about pH 3) is than water (pH 7), or look up the sound level of a lawn mower and compare it with conversation. If your teen does the red cabbage activity, make sure the knife and kettle are used with care (help if needed), and that only the listed kitchen liquids are tested.

Common difficulties:

Treating the scales as linear: pH 4 is not "twice as acidic" as pH 8; it is 10 000 times.
Forgetting the negative sign in pH = −log [H⁺].
Adding decibels directly (70 dB + 70 dB is about 73 dB, not 140 dB).
Mixing up amplitude (10 times per magnitude unit) and energy (about 32 times per unit).