What a logarithm is
A logarithm answers the question: what power do I need? The base-10 logarithm of 1000 is 3, because 10³ = 1000. Written out, log₁₀(1000) = 3.
A few easy values:
- log₁₀(1) = 0, since 10⁰ = 1
- log₁₀(10) = 1
- log₁₀(100) = 2
- log₁₀(1000) = 3
- log₁₀(0.1) = −1, since 10⁻¹ = 0.1
Logarithms of numbers that are not neat powers of ten are found with a calculator or software. One worth remembering is log₁₀(2) ≈ 0.301, which leads to a famous rule of thumb below.
The useful property
Logarithms turn multiplication into addition: log(a × b) = log(a) + log(b). For example, log₁₀(100 × 1000) = log₁₀(100,000) = 5, and 2 + 3 = 5.
This matters when quantities span many orders of magnitude. Human hearing, radio signal strength and amplifier gains can vary by factors of millions. A log scale compresses that range into manageable numbers, much as scientific notation does for writing big and small values.
The decibel
The decibel scale is built on this idea. A decibel (dB) is not a unit of a physical quantity. It expresses a ratio between two values on a logarithmic scale. For power:
dB = 10 × log₁₀(P₂ / P₁)
Some power ratios:
| Power ratio | Calculation | Result |
|---|---|---|
| 1 | 10 × log₁₀(1) | 0 dB |
| 10 | 10 × 1 | 10 dB |
| 100 | 10 × 2 | 20 dB |
| 1000 | 10 × 3 | 30 dB |
| 2 | 10 × 0.301 | about 3 dB |
| 0.5 | 10 × (−0.301) | about −3 dB |
| 0.1 | 10 × (−1) | −10 dB |
Positive decibels mean gain, negative decibels mean loss, and 0 dB means no change. Doubling power is about +3 dB, and halving it is about −3 dB.
Voltage and amplitude ratios use 20
Power is often proportional to the square of voltage (or amplitude) when the impedance stays the same. Since log(V²) = 2 × log(V), the formula for voltage becomes:
dB = 20 × log₁₀(V₂ / V₁)
Examples:
- A voltage ratio of 10 gives 20 × 1 = 20 dB. The matching power ratio is 10² = 100, and 10 × log₁₀(100) = 20 dB. Both agree.
- A voltage ratio of 2 gives 20 × 0.301 ≈ 6 dB. The power ratio is 4, and 10 × log₁₀(4) = 10 × 0.602 ≈ 6 dB.
The “20” applies only when comparing quantities that behave like voltage, pressure or amplitude, with the same impedance or conditions at both points. Mixing up 10 and 20 is a common error.
Why decibels are convenient
Because logs turn products into sums, gains in a chain simply add. Suppose an amplifier provides 20 dB of gain and a following attenuator provides −6 dB.
Net = 20 + (−6) = 14 dB.
Check it using ratios. The amplifier multiplies power by 100. An attenuation of about 6 dB multiplies power by about 0.25, since 10 × log₁₀(0.25) ≈ −6.02. The combined factor is 100 × 0.25 = 25, and 10 × log₁₀(25) ≈ 13.98, which rounds to 14 dB. The two methods agree.
Decibels with a reference
A bare dB is a ratio. When a fixed reference is attached, it becomes an absolute level. In radio and audio work, dBm expresses power relative to 1 milliwatt, so 0 dBm is 1 mW, 10 dBm is 10 mW and 30 dBm is 1 W. Sound levels in air are quoted in decibels relative to a standard reference pressure, and other suffixes exist for other references. Always check which reference a given dB figure uses, and never add or compare levels with different references without converting.
Rules of thumb
- +3 dB is roughly double the power; −3 dB is roughly half.
- +10 dB is ten times the power; +20 dB is a hundred times.
- +6 dB is roughly double the voltage; +20 dB is ten times the voltage.
These are approximations. The exact value for a doubling is 3.01 dB, which is why rough sums can drift when many steps are added.
Logarithmic axes
Charts often use a logarithmic axis, for example frequency from 10 Hz to 100 kHz. Equal distances on that axis represent equal ratios, not equal differences. On such a plot, a straight line does not mean steady additive change. When reading values off a log chart, the straight-line method of linear interpolation does not apply directly unless you interpolate on the logarithm. Frequency-response curves in datasheet graphs commonly use these axes.
Cautions
Practise with simple ratios first, check results with a second method, and keep SI prefixes straight, since milliwatts and watts differ by a factor of 1000, or 30 dB. These are educational examples. For design work, follow the definitions, standards and measurement practices used in your industry, and rely on qualified people for anything safety-related.