Logarithmic scales

What you will learn

understand what a logarithmic (log) scale is and why it is used,
interpret orders of magnitude and compare vastly different quantities,
read and interpret real-world log scales including Richter, decibel, and pH,
decide when a log scale is more appropriate than a linear scale.

Worked example 0 Real-world example: comparing earthquakes

The 2011 Tohoku earthquake (Japan) measured $9.1$ on the Richter scale. A minor tremor measures $3.1$ . How many times more energy did the Tohoku earthquake release?

The Richter scale is logarithmic: each whole-number step represents roughly $31.6$ times more energy.
Difference in magnitude: $9.1 - 3.1 = 6.0$ steps.
Energy ratio $\approx 31.6^{6} \approx 1\,000\,000\,000$ (about one billion times more energy).

Key idea: a small numerical difference on a log scale corresponds to a huge multiplicative difference in the actual quantity.

1. What is a logarithmic scale?

On a linear scale, equal steps correspond to equal additions (e.g. $0, 10, 20, 30, \ldots$ ). On a logarithmic scale, equal steps correspond to equal multiplications — typically by $10$ .

Logarithm definition

\log_{10} x = n \quad\Longleftrightarrow\quad 10^n = x

If $x$ is multiplied by $10$ , its $\log_{10}$ increases by $1$ .

A linear scale vs a log scale. On the log scale, the distance from 1 to 10 equals the distance from 10 to 100.

Orders of magnitude: we say two quantities differ by $n$ orders of magnitude if one is roughly $10^n$ times the other. For example, $100$ and $1\,000\,000$ differ by $4$ orders of magnitude because $10^4 = 10\,000$ and $100 \times 10\,000 = 1\,000\,000$ .

Worked example 1 Orders of magnitude

The mass of an ant is about $0.001$ g. The mass of an elephant is about $5\,000\,000$ g. How many orders of magnitude apart are they?

Ratio: $\dfrac{5\,000\,000}{0.001} = 5 \times 10^{9}$ .
$\log_{10}(5 \times 10^{9}) \approx 9.7$ .
The masses are roughly $10$ orders of magnitude apart.

2. Real-world logarithmic scales

The Richter scale (earthquakes)

Each whole-number increase corresponds to a tenfold increase in measured wave amplitude and roughly $31.6$ times more energy released.

Worked example 2 Richter scale comparison

Earthquake A has magnitude $5.0$ and Earthquake B has magnitude $7.0$ . Compare wave amplitudes and energy.

Amplitude ratio: $10^{7.0 - 5.0} = 10^2 = 100$ times larger.
Energy ratio: $31.6^2 \approx 1000$ times more energy.

The decibel scale (sound)

Sound intensity level in decibels (dB) is defined as:

Decibel formula

L = 10 \log_{10}\!\left(\frac{I}{I_0}\right)

where $I_0 = 10^{-12}$ W/m $^2$ is the threshold of hearing.

A $10$ dB increase means the sound intensity is $10$ times greater. A $20$ dB increase means $100$ times greater.

Worked example 3 Decibel comparison

Normal conversation is about $60$ dB. A rock concert is about $110$ dB. How many times more intense is the concert?

Difference: $110 - 60 = 50$ dB.
Intensity ratio: $10^{50/10} = 10^5 = 100\,000$ times more intense.

The pH scale (chemistry)

pH definition

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

where $[\text{H}^+]$ is the hydrogen-ion concentration in mol/L.

A decrease of $1$ pH unit means a tenfold increase in acidity (hydrogen-ion concentration).

Worked example 4 pH comparison

Lemon juice has pH $\approx 2$ and milk has pH $\approx 6.5$ . How many times more acidic is lemon juice?

pH difference: $6.5 - 2 = 4.5$ .
Acidity ratio: $10^{4.5} \approx 31\,623$ times more acidic.

3. When to use a log scale vs a linear scale

Feature	Linear scale	Log scale
Spacing	Equal differences	Equal ratios
Best for	Data in a narrow range	Data spanning many orders of magnitude
Zeroes	Can display zero	Cannot display zero ( $\log 0$ is undefined)
Negative values	Yes	No (log of negatives is undefined)
Pattern revealed	Additive trends (straight lines)	Multiplicative/exponential trends (straight lines)

Worked example 5 Choosing the right scale

A scientist measures bacteria counts at $1$ -hour intervals: $50$ , $150$ , $450$ , $1350$ , $4050$ . Should she use a linear or log scale?

The values increase by a factor of $3$ each hour (exponential growth).
On a linear scale the early values cluster near zero and the last value dominates.
On a log scale the points form a straight line, making the constant growth rate obvious.

A log scale is the better choice here.

Key formulas

Logarithm definition

\log_{10} x = n \iff 10^n = x

Decibel level

L = 10\log_{10}\!\left(\frac{I}{I_0}\right)

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

Practice

Fluency

Tier 1: basic skills

Evaluate without a calculator: $\log_{10} 1000$ .
Evaluate: $\log_{10} 0.01$ .
If $\log_{10} x = 4$ , find $x$ .
How many orders of magnitude separate $500$ from $5\,000\,000$ ?
A sound of $30$ dB is how many times more intense than the threshold of hearing ( $0$ dB)?
An earthquake of magnitude $6.0$ has wave amplitudes how many times larger than one of magnitude $4.0$ ?
A solution has $[\text{H}^+] = 10^{-3}$ mol/L. Find its pH.
If the pH drops from $7$ to $5$ , by what factor has the hydrogen-ion concentration increased?

Reasoning

Tier 2: mixed practice

Two earthquakes measure $4.5$ and $6.5$ on the Richter scale. (a) Compare their wave amplitudes. (b) Estimate the energy ratio.
A vacuum cleaner produces $75$ dB and a whisper is $20$ dB. How many times more intense is the vacuum cleaner?
A scientist records data points $2, 20, 200, 2000, 20\,000$ . Explain why a log scale is more suitable for graphing this data.
Coffee has pH $5$ and household ammonia has pH $11.5$ . Which is more acidic, and by what factor of hydrogen-ion concentration?
The population of a town doubles every $10$ years. If the current population is $5000$ , calculate the population after $50$ years and explain why a log-scale graph of this growth would appear as a straight line.
On a log-scaled graph, two data points appear $3$ cm apart and each centimetre represents one order of magnitude. What is the ratio of the larger value to the smaller?

Reasoning

Tier 3: explain and apply

Explain in your own words why $\log_{10} 0$ is undefined and what this means for graphing on a log scale.
The apparent magnitude scale for stars decreases by $1$ for each factor of $\approx 2.512$ increase in brightness. A star of magnitude $1$ is how many times brighter than a star of magnitude $6$ ? Show your working.
A student says “an earthquake of magnitude $8$ is twice as strong as one of magnitude $4$ .” Explain why this statement is incorrect and calculate the actual amplitude ratio.
Create a table listing five quantities from everyday life that span at least $8$ orders of magnitude (e.g. mass, distance, or time). Explain why a log scale is useful for displaying them together.

Challenge

Reasoning

Harder reasoning

The energy $E$ (in joules) released by an earthquake of Richter magnitude $M$ is approximately $\log_{10} E = 1.5M + 4.8$ . Find the energy released by earthquakes of magnitude $5.0$ and $8.0$ , and verify that the ratio is approximately $31.6^3$ .
Two sound sources produce $80$ dB and $80$ dB respectively. When played simultaneously, the total intensity doubles but the combined level is not $160$ dB. Find the actual combined decibel level using the formula $L = 10\log_{10}(I_1/I_0 + I_2/I_0)$ .
A culture of bacteria grows from $100$ to $100\,000\,000$ in $24$ hours at a constant rate. (a) How many orders of magnitude of growth is this? (b) If you plot the count on a log scale against time, what shape will the graph be? (c) Find the hourly growth factor.
The Moment Magnitude Scale (used for large earthquakes) is defined by $M_w = \frac{2}{3}\log_{10}(M_0) - 6.07$ , where $M_0$ is the seismic moment in Nm. If $M_0$ increases by a factor of $1000$ , by how much does $M_w$ increase?