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Logarithmic Scales: pH, Decibels & Magnitude

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A scale that counts factors, not amounts

Hydrogen ion concentrations in ordinary solutions range from about 1 to 0.00000000000001 mol/L — fourteen orders of magnitude. No linear axis can display that. A logarithmic scale replaces the quantity by its exponent, so each unit step means a fixed factor rather than a fixed amount. pH, decibels and earthquake magnitude are all this device applied to different quantities.

Three logarithmic scales
pH = −log[H⁺] · dB = 10·log(I/I₀) · magnitude difference: I₁/I₂ = 10^(M₁ − M₂)
The minus sign in pH exists so that ordinary concentrations give positive pH values. I₀ for decibels is the threshold of hearing, 10⁻¹² W/m².

One unit means one factor

Because the scale is built from a logarithm, a difference in scale values corresponds to a ratio in the underlying quantity. One pH unit is a factor of 10 in [H⁺] — so pH 4 is ten times as acidic as pH 5, and pH 3 is a hundred times as acidic as pH 5. One magnitude unit is a factor of 10 in seismic amplitude. Decibels carry a factor of 10 in the definition, so it takes ten decibels to multiply intensity by ten, and 3 dB is roughly a doubling.

Worked example

A solution has [H⁺] = 3.2 × 10⁻⁵ mol/L. Find its pH, and find the concentration of a solution with pH 8.4.

  1. 1.pH = −log(3.2 × 10⁻⁵) = −[log 3.2 + log 10⁻⁵] = −[0.5051 − 5] = 4.495.
  2. 2.So pH ≈ 4.49 — acidic, as expected for a concentration well above the neutral 10⁻⁷.
  3. 3.For pH 8.4, invert the definition: [H⁺] = 10^(−pH) = 10^(−8.4).
  4. 4.10^(−8.4) = 10^(0.6) × 10^(−9) ≈ 3.98 × 10⁻⁹ mol/L.
Answer: pH ≈ 4.49, and pH 8.4 corresponds to [H⁺] ≈ 4.0 × 10⁻⁹ mol/L. The second solution is basic, and the pH gap of about 3.9 units means the first is roughly 10^3.9 ≈ 8,000 times as acidic.
Checkpoint

How much more acidic is a solution of pH 3 than one of pH 6?

Watch out

Decibels are not additive in intensity. Two identical 70 dB sources together give about 73 dB, not 140 dB — doubling the intensity adds 10·log 2 ≈ 3 dB. Adding decibel values directly is meaningless.

Worked example

A sound has intensity 4.5 × 10⁻⁶ W/m². Find its decibel level, and find how many times more intense a 95 dB sound is than a 75 dB sound.

  1. 1.dB = 10·log(I/I₀) with I₀ = 10⁻¹²: the ratio is 4.5 × 10⁻⁶/10⁻¹² = 4.5 × 10⁶.
  2. 2.log(4.5 × 10⁶) = log 4.5 + 6 ≈ 0.6532 + 6 = 6.6532.
  3. 3.dB = 10(6.6532) ≈ 66.5 dB.
  4. 4.For the comparison: a 20 dB difference means 20 = 10·log(I₁/I₂), so log(I₁/I₂) = 2.
  5. 5.Therefore I₁/I₂ = 10² = 100.
Answer: The sound is about 66.5 dB, and 95 dB is 100 times as intense as 75 dB. The factor of 10 in the decibel definition means you divide the decibel difference by 10 before exponentiating — which is why 20 dB is a hundredfold, not a twentyfold.

Why the ear and the eye agree with the logarithm

Human perception of loudness and brightness is roughly logarithmic: equal ratios of intensity feel like equal steps. Doubling the power of a speaker does not sound twice as loud. So a decibel scale is not merely a convenience for cramming a wide range onto one axis — it happens to align with how the sensation actually scales, which is why sound is quoted in decibels rather than watts per square meter.

Checkpoint

An earthquake of magnitude 7.0 releases how much more seismic wave amplitude than one of magnitude 5.0?

Answer the 2 checkpoints as you read.

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