A logarithmic scale is useful when quantities cover a very large range. Instead of measuring the quantity
directly, the scale measures the logarithm of a ratio or concentration.
1. Main idea
A logarithm turns multiplication into addition.
\[
\log_{10}(ab)=\log_{10}(a)+\log_{10}(b)
\]
This means a fixed step on a logarithmic scale represents multiplication by a fixed factor in the
real-world quantity.
2. Intensity decibel scale
Sound intensity level is often measured using decibels:
\[
L=10\log_{10}\left(\frac{I}{I_0}\right)
\]
Here, \(I\) is the sound intensity and \(I_0\) is a reference intensity.
The ratio \(I/I_0\) must be positive.
To reverse the formula:
\[
\frac{I}{I_0}=10^{L/10}
\]
An increase of \(10\text{ dB}\) means the intensity ratio is multiplied by \(10\).
3. Worked decibel example
Suppose the sound intensity is \(10000\) times the reference intensity.
\[
\frac{I}{I_0}=10000
\]
Substitute into the decibel formula:
\[
L=10\log_{10}(10000)
\]
Since \(10000=10^4\), we get:
\[
\log_{10}(10000)=4
\]
Therefore:
\[
L=10\cdot4=40
\]
So the sound level is:
\[
\boxed{40\text{ dB}}
\]
4. Amplitude and pressure decibels
When decibels compare amplitude or pressure ratios, the factor is usually \(20\), not \(10\):
\[
L=20\log_{10}\left(\frac{A}{A_0}\right)
\]
To reverse the formula:
\[
\frac{A}{A_0}=10^{L/20}
\]
This happens because intensity is proportional to the square of many amplitude-like quantities.
5. pH scale
The pH scale measures hydrogen ion concentration:
\[
\text{pH}=-\log_{10}[H^+]
\]
To reverse the formula:
\[
[H^+]=10^{-\text{pH}}
\]
Because of the negative sign, a lower pH means a higher hydrogen ion concentration.
A change of \(1\) pH unit means a factor of \(10\) change in \([H^+]\).
6. Worked pH example
Suppose:
\[
[H^+]=10^{-6}
\]
Then:
\[
\text{pH}=-\log_{10}(10^{-6})
\]
Since \(\log_{10}(10^{-6})=-6\), we get:
\[
\text{pH}=6
\]
7. Richter amplitude scale
A simplified Richter-style comparison measures earthquake amplitude ratios using:
\[
\Delta M=\log_{10}\left(\frac{A}{A_0}\right)
\]
To reverse the formula:
\[
\frac{A}{A_0}=10^{\Delta M}
\]
A magnitude difference of \(1\) means an amplitude ratio of \(10\).
A magnitude difference of \(2\) means an amplitude ratio of \(100\).
8. Approximate earthquake energy ratio
A common approximation for earthquake energy ratio is:
\[
\text{energy ratio}\approx10^{1.5\Delta M}
\]
This means a magnitude increase of \(1\) corresponds to about \(10^{1.5}\), or roughly \(31.6\), times
more energy.
9. Why logarithmic scales are powerful
Logarithmic scales compress very large ranges into manageable numbers.
- A ratio of \(10\) becomes \(\log_{10}(10)=1\).
- A ratio of \(100\) becomes \(\log_{10}(100)=2\).
- A ratio of \(1000\) becomes \(\log_{10}(1000)=3\).
- A ratio of \(10000\) becomes \(\log_{10}(10000)=4\).
This is why large physical changes can become simple scale differences.
10. Summary table
| Scale |
Formula |
Inverse formula |
Meaning of one scale unit |
| pH |
\(\text{pH}=-\log_{10}[H^+]\) |
\([H^+]=10^{-\text{pH}}\) |
A \(1\)-unit pH change means a factor of \(10\) in concentration. |
| Intensity dB |
\(L=10\log_{10}(I/I_0)\) |
\(I/I_0=10^{L/10}\) |
A \(10\text{ dB}\) change means a factor of \(10\) in intensity. |
| Amplitude dB |
\(L=20\log_{10}(A/A_0)\) |
\(A/A_0=10^{L/20}\) |
A \(20\text{ dB}\) change means a factor of \(10\) in amplitude. |
| Richter amplitude |
\(\Delta M=\log_{10}(A/A_0)\) |
\(A/A_0=10^{\Delta M}\) |
A \(1\)-unit magnitude change means a factor of \(10\) in amplitude. |
11. Common mistakes
- Use \(10\log_{10}\) for intensity ratios, but \(20\log_{10}\) for amplitude or pressure ratios.
- For pH, remember the negative sign: \(\text{pH}=-\log_{10}[H^+]\).
- Ratios and concentrations used inside logarithms must be positive.
- Do not treat equal scale differences as equal additive physical changes; they are multiplicative changes.
