1
A reading on a log scale is an exponent
pH 5 is not an amount of anything. It says the hydrogen ion concentration is 10^−5 moles per litre, and the scale reports the −5. That is why the reading moves by 1 when the quantity moves by a factor of 10.
Lesson · Exponential and Logarithmic Functions · Applications
Logarithmic Scales
Decibels, pH and earthquake magnitude are one construction used three times: report the logarithm of a ratio instead of the ratio. Equal steps on the scale then stand for equal factors on the quantity, which is what lets a range of fourteen powers of ten fit on one axis.
pH = −log[H+] one unit down = 10× more H+
dB = 10 · log(I / I₀) +10 dB = 10× intensity; 2× = +3.01 dB
M = log(A / A₀) +2 magnitude = 100× amplitude
adding on the scale = multiplying the quantity
Three scales, one construction
Core ideas
What this lesson establishes
Three definitions that differ only in a reference value and a multiplier, and one rule for the arithmetic that is legal on them.
2
Equal distances are equal ratios
On a logarithmic axis the gap from 1 to 2 is the same as the gap from 10 to 20 and from 100 to 200, because each is log 2 ≈ 0.30103. Distances add where the quantities multiply, which is the product law of “The Laws of Logarithms” drawn as a picture.
3
Differences are the only arithmetic that transfers
Two magnitudes cannot be averaged and two pH readings cannot be added, because the quantities underneath do not combine that way. What a difference means is fixed and reference-free: two magnitudes apart is 10² = 100 times the amplitude, on any seismograph.
The material
Scales that report a logarithm
Three definitions, the property they share, and what the arithmetic on them does and does not mean.
Audible sound intensities run from about 10^−12 watts per square metre at the threshold of hearing to about 10^2 at the point of damage: fourteen powers of ten. On a linear axis the quiet end of that range is invisible. Reporting the exponent instead of the quantity turns the whole range into the interval from 0 to 140.
Definition · Decibel
L = 10 · log(I / I₀), where I is intensity and I₀ = 10^−12 watts per square metre is a fixed reference. L carries no units: it is ten times the exponent of a ratio. The factor of ten is arbitrary — the unit is one tenth of a bel, and the bel alone was too coarse to be useful.
Definition · pH
pH = −log[H+], where [H+] is the hydrogen ion concentration in moles per litre. The minus sign exists so that ordinary solutions get positive readings: [H+] = 10^−7 gives pH 7. It also reverses the direction, so a lower pH means more hydrogen ions.
Definition · Magnitude
M = log(A / A₀), where A is the amplitude recorded and A₀ a fixed reference amplitude. Magnitude 6 is ten times the amplitude of magnitude 5. The energy released is separately found to scale as 10^(1.5 · M), which is an empirical relation rather than part of the definition.
All three have the shape S = k · log(Q / Q₀). The reference Q₀ makes the argument a pure number, which it must be before a logarithm can touch it. The multiplier k sets the size of one step: 1 for magnitude, 10 for decibels, −1 for pH.
Theorem · A difference on the scale is a ratio of quantities
If S = k · log(Q / Q₀) then S₂ − S₁ = k · log(Q₂ / Q₁), and so Q₂ / Q₁ = 10^((S₂ − S₁) / k).
The reference cancels, by the quotient law of “The Laws of Logarithms”. This is why two readings can be compared by anyone, while a single reading means nothing until the reference is stated.
decibels, k = 10
+10 dB → 10^1 = 10× the intensity
+30 dB → 10^3 = 1000× the intensity
doubling the intensity → 10 · log 2 ≈ 3.0103 dB
pH, k = −1
pH 6 down to pH 3 → 10^((3 − 6)/(−1)) = 10^3 = 1000× the [H+]
magnitude, k = 1
5.0 up to 7.0 → 10^2 = 100× the amplitude
10^(1.5 · 2) = 1000× the energy
Why the sum of two sources is not double the reading
Two identical machines each measured at 60 dB run together. The intensities add, so the total is 2I, and the level is 10 · log(2I / I₀) = 60 + 10 · log 2 ≈ 63.01 dB. Ten identical machines give exactly 70 dB, since 10 · log 10 = 10. Adding the readings instead — 60 + 60 = 120 — would correspond to multiplying the intensities, which is what a sum of logarithms means.
A caution about the word loud. An increase of 10 dB is ten times the intensity and is usually reported as sounding about twice as loud. Perceived loudness is a psychological measurement with its own scale; everything in this lesson is about intensity, which is the quantity actually inside the logarithm.
Reading a logarithmic axis
On a log axis the gridlines within each decade are unevenly spaced and the spacing repeats. Between 1 and 10, the mark for 2 sits 30.1% of the way along and the mark for 5 sits 69.9% along, because log 2 ≈ 0.30103 and log 5 ≈ 0.69897. The identical pattern appears between 10 and 100, and between 100 and 1000.
One consequence is worth carrying away. If y = a · b^x, then log y = log a + x · log b, which is a linear function of x. So an exponential plotted with a logarithmic vertical axis is a straight line, and its slope is log b. A straight line on such a plot is evidence of a constant ratio per step, which is exactly the definition of exponential from “Exponential Functions”.
Example · A number you can check
Normal conversation is about 60 dB and a jet engine at close range about 140 dB. The gap of 80 on the scale is a factor of 10^8 in intensity: a hundred million. That compression is the whole point of the construction, and it is also why the gap sounds much smaller than it is.
Interactive
Scales that are logarithms
Decibels, pH and magnitude
Method
Comparing two readings on a log scale
Four steps. The reference value is never needed, and that is the point of step two.
1
Identify the multiplier in front of the logarithm
1 for magnitude, 10 for decibels, −1 for pH. Everything else in the comparison follows from that number.
2
Subtract the two readings
The difference is what carries the information. The reference cancels out of it by the quotient law, so nothing about I₀ or A₀ needs to be known.
3
Divide by the multiplier and raise 10 to the result
Q₂ / Q₁ = 10^((S₂ − S₁) / k). For pH the negative k flips the direction, which is the arithmetic behind "lower pH, more acid".
4
Report a factor, not a difference
The answer is "a thousand times the concentration", not "three more acidic". If the answer still has the units of the scale on it, the conversion has not happened yet.
Steps and factors
Score 0 / 0Choose the answer you can justify.
Four comparisons, one method
Each one is a subtraction, a division by k, and a power of ten.
(1) 60 dB up to 90 dB k = 10
difference 30 ratio = 10^(30/10) = 10^3
the intensity is 1000 times as large
(2) pH 6 down to pH 3 k = −1
difference −3 ratio = 10^(−3 / −1) = 10^3
1000 times the [H+]; pH 3 is the more acidic
(3) magnitude 5.0 up to 7.0 k = 1
difference 2 amplitude ratio = 10^2 = 100
energy ratio = 10^(1.5 · 2) = 1000
(4) two 60 dB sources together
the intensities add: 2I
new level = 60 + 10 · log 2 ≈ 63.01 dB not 120
None of the first three used a reference value. I₀, A₀ and the mole never appeared, because k · log(Q₂/Q₀) − k · log(Q₁/Q₀) = k · log(Q₂/Q₁) — the quotient law removing the thing both readings had in common.
Line (4) is the one that surprises people. Two identical sources are three decibels up, not double the reading and not double anything on the scale. Ten identical sources are exactly ten decibels up. The quantities double and decuple; the readings, which are logarithms, add log 2 and log 10 worth of steps.
For a faded scale comparison, take a sound level rising from 72 dB to 92 dB. The difference 20 and multiplier k = 10 are supplied. Finish I₂/I₁ = 10^(20/10), report the result as an intensity factor, and explain why neither 20 nor 92/72 answers the question. Check the direction: the larger reading must represent the larger intensity.
Common mistakes
1
Reading a difference as a factorMagnitude 7 is not "two more" than magnitude 5 in any physical sense, and it is certainly not twice as large. It is 100 times the amplitude and about 1000 times the energy. The reading is an exponent, and the exponent is the thing that went up by two.
2
Adding readings when the quantities addTwo 60 dB sources make about 63 dB, not 120. Adding logarithms multiplies the quantities underneath, so 60 + 60 describes the ratio (I₁/I₀) · (I₂/I₀) = 10^12, a product of two ratios rather than a sum of two intensities. Add the intensities first, then take the logarithm once.
3
Reading pH in the intuitive directionThe minus sign makes the scale run backwards against intuition: pH 2 is more acidic than pH 6, and a higher pH means fewer hydrogen ions. There is no way to feel your way to this. Write the definition down and follow it.
Check your understanding
✓
Finish when you turn a difference into a factor without looking anything up.
Given two readings on any of the three scales, you should be able to name the multiplier k, subtract, and report 10^((S₂ − S₁)/k) as a factor on the quantity. If the answer comes out carrying the units of the scale, the conversion has not been done.
Note: This is the last lesson of the course, and it is the definition in “What a Logarithm Is” doing ordinary work: every reading on these scales is an exponent, and every legal manipulation of them is one of the three laws of “The Laws of Logarithms” read in the other direction. Sequences and Series picks up the sequences that these constant-ratio quantities have been all along.