How this instrument works
The decibel measures a power ratio, not a power. Take the power you measured, divide it by a reference power, and multiply the base-ten logarithm of that ratio by ten: dB = 10·log₁₀(P₂ ⁄ P₁). A telephone line that halves the signal loses about 3 dB; an amplifier that outputs ten times what it takes in gains exactly 10 dB, because log₁₀(10) is exactly 1. The logarithm is what makes the unit useful — a hearing range that spans a trillion-to-one in raw power collapses to a scale running from roughly 0 to 120.
Bell Telephone Laboratories introduced the unit in the 1920s to track how much signal a phone line lost over distance, adding losses across repeater stations the way you would add lengths of cable. The bel itself is log₁₀(P₂⁄P₁); the decibel is a tenth of a bel, which is why the formula carries a factor of ten rather than one. Engineers kept it because logarithmic addition matches how cascaded gains and losses actually combine — and, separately, because human hearing responds to loudness in a roughly logarithmic way too.
The formula breaks down at its edges. If the measured power is zero, log₁₀(0) is undefined — physically there is no power ratio to describe, and the instrument will not return a finite figure. If the measured power is smaller than the reference, the result is negative: a loss or attenuation, not an error. And because P₂⁄P₁ is dimensionless, both powers must already share the same unit — watts against watts, milliwatts against milliwatts — or the ratio, and everything computed from it, is meaningless.
- Enter the baseline figure in Reference power — the power level you are comparing against, such as an amplifier's input or a fixed standard like 1 mW.
- Enter the power you actually measured in Measured power, using the same unit you used for the reference power.
- Read the result in Level, dB. A positive number is a gain over the reference; a negative number is a loss.
- To sanity-check a doubling of power, set Reference power to 1 and Measured power to 2; the readout should land near 3.01 dB.
Worked example — a tenfold power gain
An audio engineer bench-tests a preamp by feeding it exactly 1 watt and measuring 10 watts at the output. With Reference power at 1 and Measured power at 10, the formula gives dB = 10·log₁₀(10 ⁄ 1) = 10·log₁₀(10) = 10·1 = 10 dB. Because log₁₀(10) is exactly 1 by definition of a base-ten logarithm, this is one of the rare cases where the decibel figure comes out as a clean integer rather than a rounded decimal.
That exactness is what makes 10:1 the benchmark ratio for the whole scale: every additional factor of ten in the power ratio adds another clean 10 dB, so a 100:1 ratio reads 20 dB and a 1000:1 ratio reads 30 dB. Compare that to doubling the power, which only adds about 3.01 dB, since log₁₀(2) is an irrational 0.30103 rather than a round number — the reason engineers memorize '3 dB up' instead of calculating it fresh each time.
Questions
Why does the decibel formula multiply by 10 instead of 1?
Because a decibel is a tenth of a bel, the original unit Bell Labs defined as log₁₀(P₂⁄P₁). Multiplying that logarithm by 10 converts bels to decibels, giving finer resolution — whole numbers instead of awkward tenths — while keeping the underlying arithmetic identical to log₁₀(P₂⁄P₁).
What does a negative dB value mean?
It means the measured power is smaller than the reference — a loss rather than a gain. A cable that delivers 0.5 W from a 1 W input returns 10·log₁₀(0.5) = −3.01 dB. Negative results are normal for attenuators, long cable runs, or any lossy component; nothing about the calculation has gone wrong.
Do the two power values need to be in the same unit?
Yes. The formula only cares about the ratio P₂⁄P₁, so watts must be compared against watts and milliwatts against milliwatts. Mix watts with milliwatts and the ratio — and every decibel figure built from it — is off by a factor of 1000, roughly a 30 dB error.
Why is 3 dB often called 'double the power'?
Because log₁₀(2) is about 0.301, so 10·log₁₀(2) is about 3.01 dB — close enough to 3 dB that engineers round it and use '+3 dB' as shorthand for doubling power, and '−3 dB' for halving it. It is an approximation, not an exact identity like the clean 10 dB you get from a tenfold ratio.
What happens if the measured power is zero?
The formula returns no finite answer, because log₁₀(0) is undefined — there is no power ratio for a signal that is not there. Physically this corresponds to negative infinity dB, the theoretical floor of the scale; a real reading of exactly zero usually points to a broken connection rather than a figure worth expressing in decibels.
Is dB the same thing as dBm or dBW?
No. Plain dB expresses a ratio between two powers you supply — there is no absolute value attached. dBm and dBW use the same log-ratio formula with the reference power fixed at 1 milliwatt or 1 watt respectively, so they report an absolute power level instead of a comparison between two figures you choose.