9.6: Logarithmic Scales
- Page ID
- 132164
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Linear axes work well when data spans a modest range. When data spans multiple orders of magnitude — which happens constantly in electrical engineering — logarithmic scales become necessary.
On a log scale, each decade (factor of 10) takes the same physical space. Values are spaced as: 0.001, 0.01, 0.1, 1, 10, 100 ... rather than 0, 1, 2, 3, 4 ...
| Situation | Why log scale helps |
|---|---|
| Frequency response plots (Bode plots) | Frequency ranges from ~1 Hz to ~1 MHz — linear scale crushes the low end |
| Component value selection | Capacitors span pF (\(10^{-12}\)) to mF (\(10^{-3}\)) — 9 orders of magnitude |
| Signal amplitudes in dB | Decibels are a logarithmic unit by definition |
| Reaction rates, material fatigue cycles | Quantities that span many powers of 10 |
A common mistake: plotting data spanning multiple orders of magnitude on a linear scale, then concluding that small values are negligible because they appear flat near the axis. On a log scale, those same values may show important structure.
The two graphs below use the same RC low-pass frequency response data. The first graph uses a linear frequency axis, which compresses the low-frequency region. The second graph uses a logarithmic frequency axis, which spreads each decade evenly and makes the full response easier to interpret.
The linear frequency axis places most of the visual space at high frequencies. When the data spans several orders of magnitude, important low-frequency behavior is crowded near the left edge.
Figure \(\PageIndex{1}\): RC low-pass frequency response plotted with a linear frequency axis. Because the frequency range spans several orders of magnitude, the low-frequency portion of the response is compressed near the left side of the graph. (Copyright; Jonathan Compton, CC BY-NC 4.0).
The logarithmic frequency axis gives equal visual spacing to each decade. This makes the transition region and overall trend easier to read across the full frequency range.
Figure \(\PageIndex{2}\): The same RC low-pass frequency response plotted with a logarithmic frequency axis. Equal spacing by decade makes the full response visible, including the transition region and high-frequency roll-off. (Copyright; Jonathan Compton, CC BY-NC 4.0).