Logarithmic Scales: pH, Decibels, Richter

A logarithmic scale spaces numbers by powers of 10, so each equal step on the scale multiplies the quantity by 10. pH, decibels and earthquake magnitude all use log scales. A difference of $d$ units on a base-10 log scale means one quantity is $10^d$ times the other, so pH 3 is $10^3 = 1000$ times more acidic than pH 6.

Before this lesson: You should know Logarithms, Properties of Logarithms and Scientific Notation.

Why Use a Logarithmic Scale?

Some quantities cover a huge range. The sound of a whisper and the sound of a jet engine differ in intensity by a factor of about 100 billion. On an ordinary (linear) scale, the small values would crowd into a spot too tiny to see.

A log scale fixes that. It records the exponent instead of the number, so 10 becomes 1, 1,000 becomes 3 and 1,000,000 becomes 6. Each step of 1 on the scale means “10 times as much.”

Two number lines from 1 to 1,000,000. The top, solid blue logarithmic scale has equally spaced marks at 1, 10, 100, 1,000, 10,000, 100,000 and 1,000,000, so each equal step multiplies by 10. The bottom, dashed orange linear scale has marks every 250,000, and a green dot shows that all the numbers from 1 to 10,000 crowd into a tiny piece at the left end. — On a log scale, each equal step multiplies by 10. On a linear scale, small values crowd together at one end.

Common Logarithmic Scales

Scale	Formula	What one unit means
pH (acidity)	$\text{pH} = -\log[\text{H}^+]$	One pH unit lower is 10 times more acidic
Decibels (sound)	$L = 10 \log \frac{I}{I_0}$ , with $I_0 = 10^{-12}$ W/m²	10 dB louder is 10 times the intensity
Richter magnitude (earthquakes)	$M = \log \frac{A}{A_0}$	One unit higher is 10 times the wave amplitude

Here $[\text{H}^+]$ is the hydrogen ion concentration in moles per liter, $I$ is sound intensity, and $A$ is the amplitude of the seismic wave. $I_0$ and $A_0$ are reference values.

The comparison rule

\text{If two values differ by } d \text{ on a base-10 log scale, their quantities differ by a factor of } 10^d.

For decibels, divide the difference by 10 first, because of the 10 in front of the log: a difference of $d$ dB is a factor of $10^{d/10}$ .

Worked Examples

Use a calculator for logs that are not whole numbers. Round to the nearest hundredth.

Example 1: Find a pH

Black coffee has $[\text{H}^+] = 3.2 \times 10^{-5}$ moles per liter. Find its pH.

\text{pH} = -\log(3.2 \times 10^{-5})

Write the formula.

\log(3.2 \times 10^{-5}) = \log 3.2 + \log 10^{-5}

Use the product rule (see properties of logarithms)

Simplify: $\log 10^{-5} = -5$ and $\log 3.2 \approx 0.505$ , so the log is about $0.505 - 5 = -4.495$

\text{pH} \approx 4.49

Change the sign.

Answer: about 4.49, so coffee is acidic (below 7).

Example 2: Compare two pH values

Lemon juice has a pH of about 2. Milk has a pH of about 6. How many times more acidic is lemon juice?

6 - 2 = 4

Find the difference on the scale.

Each pH unit is a factor of 10, so the factor is $10^4$

10^4 = 10{,}000

Evaluate.

Answer: lemon juice is about 10,000 times more acidic. A small-looking difference of 4 hides a huge ratio.

Example 3: Find a decibel level

A busy street has sound intensity $I = 10^{-5}$ W/m². Find its loudness in decibels.

L = 10 \log \frac{10^{-5}}{10^{-12}}

Write the formula.

\frac{10^{-5}}{10^{-12}} = 10^{-5 - (-12)} = 10^{7}

Divide the powers by subtracting exponents (see exponent rules)

\log 10^7 = 7

Take the log.

L = 10 \cdot 7 = 70

Multiply by 10.

Answer: 70 dB

Example 4: Compare two sounds

A concert is 95 dB, and a vacuum cleaner is 75 dB. How many times more intense is the concert?

95 - 75 = 20

Find the difference in decibels.

\frac{20}{10} = 2

Divide by 10 for decibels.

The factor is $10^2 = 100$

Answer: the concert is 100 times as intense, even though 95 is not much bigger than 75.

Example 5: Word problem with earthquakes

An earthquake of magnitude 7.2 hit one region, and an earthquake of magnitude 5.4 hit another. How many times larger was the seismic wave amplitude of the bigger quake?

7.2 - 5.4 = 1.8

Find the difference in magnitude.

Each unit is a factor of 10 in amplitude: the factor is $10^{1.8}$

10^{1.8} \approx 63.10

Evaluate with a calculator.

Answer: about 63 times larger. The released energy differs by even more, since energy grows faster than amplitude.

Common Mistakes

Mistake	Example of the error	Fix
Treating the scale as linear	pH 2 is “3 times” as acidic as pH 6	A difference of 4 means a factor of $10^4 = 10{,}000$ .
Forgetting the 10 in decibels	A 20 dB difference is a factor of $10^{20}$	Divide by 10 first: $10^{20/10} = 100$ .
Dropping the minus sign in pH	$\text{pH} = \log(3.2 \times 10^{-5}) \approx -4.49$	pH uses $-\log$ , so the answer is positive: 4.49.
Getting the direction wrong	Thinking a higher pH is more acidic	Lower pH means more $\text{H}^+$ , so more acidic.
Using ln instead of log	$-\ln(10^{-7}) \approx 16.1$	These scales use base-10 logs: $-\log(10^{-7}) = 7$ .

Related Methods

Going backward

To find the quantity from a scale value, undo the log with a power of 10. For pH 8.3, $[\text{H}^+] = 10^{-8.3} \approx 5.01 \times 10^{-9}$ moles per liter. This is the same skill as in solving logarithmic equations.

Log axes on graphs

Graphs can use a log scale on one axis. Then exponential data looks like a straight line, which is the idea behind semi-log plots.

Other bases

Musical octaves use a base-2 scale: each octave up doubles the frequency. The comparison rule is the same, with 2 in place of 10.

Practice Problems

Round answers to the nearest hundredth when needed.

Find the pH when $[\text{H}^+] = 10^{-7}$ .
Find the pH when $[\text{H}^+] = 4.5 \times 10^{-3}$ .
A solution has pH 9.2. Find $[\text{H}^+]$ .
How many times more acidic is a solution with pH 2 than one with pH 5?
Find the decibel level of a sound with $I = 10^{-3}$ W/m².
Find the decibel level of a sound with $I = 5 \times 10^{-6}$ W/m².
How many times more intense is an 80 dB sound than a 50 dB sound?
If a sound’s intensity doubles, by how many decibels does its level go up?
How many times larger is the amplitude of a magnitude 6.0 earthquake than a magnitude 4.0 earthquake?
An earthquake’s wave amplitude is 500 times that of a magnitude 3.1 quake. What is its magnitude?

Show answers

\text{pH} = -\log 10^{-7} = -(-7) = 7

Substitute into the pH formula.

Answer: 7 (neutral)

\text{pH} = -\log(4.5 \times 10^{-3}) = -(\log 4.5 - 3)

Substitute into the pH formula.

$\log 4.5 \approx 0.653$ , so $\text{pH} \approx -(0.653 - 3) = 2.347$
Answer: about 2.35

[\text{H}^+] = 10^{-9.2}

Undo the log.

10^{-9.2} \approx 6.31 \times 10^{-10}

Evaluate.

Answer: about $6.31 \times 10^{-10}$ moles per liter

5 - 2 = 3

Find the difference in pH.

10^3 = 1000

Factor.

Answer: 1,000 times more acidic

L = 10 \log \frac{10^{-3}}{10^{-12}} = 10 \log 10^{9}

Substitute into the decibel formula.

= 10 \cdot 9 = 90

Multiply.

Answer: 90 dB

L = 10 \log \frac{5 \times 10^{-6}}{10^{-12}} = 10 \log(5 \times 10^{6})

Substitute into the decibel formula.

\log(5 \times 10^6) = \log 5 + 6 \approx 0.699 + 6 = 6.699

Split the log of the product.

L \approx 10 \cdot 6.699 = 66.99

Multiply.

Answer: about 66.99 dB

80 - 50 = 30

Find the difference in decibels.

10^{30/10} = 10^3 = 1000

Factor.

Answer: 1,000 times as intense

10 \log \frac{2I}{I_0} - 10 \log \frac{I}{I_0} = 10 \log 2

New level minus old level.

10 \log 2 \approx 10(0.301) = 3.01

Evaluate.

Answer: about 3.01 dB

6.0 - 4.0 = 2

Find the difference in magnitude.

10^2 = 100

Factor.

Answer: 100 times larger

10.

The magnitudes differ by $\log 500$

\log 500 \approx 2.699

Use a calculator to find the log.

3.1 + 2.699 = 5.799

Add.

Answer: about 5.80

FAQ

What is a logarithmic scale?

It is a scale where each equal step multiplies the quantity by the same factor, usually 10. It records the exponent instead of the number itself.

How much stronger is a magnitude 6 earthquake than a magnitude 5?

Its wave amplitude is 10 times larger. The energy it releases is about 32 times larger, because energy grows faster than amplitude.

Why is pH negative log?

Hydrogen ion concentrations are tiny numbers like $10^{-7}$ . Their logs are negative, so the minus sign turns them into easy positive numbers like 7.

Is 60 dB twice as loud as 30 dB?

No. 60 dB is $10^{30/10} = 1000$ times the intensity of 30 dB. People hear it as about 8 times as loud, but the physical intensity is 1,000 times greater.

Summary

On a base-10 log scale, each step of 1 multiplies the quantity by 10.
pH: $-\log[\text{H}^+]$ ; lower pH is more acidic.
Decibels: $10 \log \frac{I}{I_0}$ ; every 10 dB is a factor of 10 in intensity.
Richter magnitude: $\log \frac{A}{A_0}$ ; each unit is 10 times the amplitude.
A difference of $d$ units means a factor of $10^d$ (or $10^{d/10}$ for decibels).

Previous lesson: Partial Fraction Decomposition

Next lesson: Semi-Log Plots

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