The decibel calculator above does the general arithmetic of the decibel scale rather than any one application of it. A decibel is not a unit of sound. It is a way of writing a ratio between two quantities on a base-ten logarithmic scale, and it is used just as heavily for radio power, amplifier gain, cable loss, optical attenuation and signal-to-noise ratios as it is for acoustics. This page handles the ratio arithmetic in all three directions people need it.
Arb Digital publishes free calculators that isolate the step where mistakes actually happen, and with decibels that step is the factor in front of the logarithm. Ten for power, twenty for amplitude — and the two rules exist because power goes as the square of amplitude, not because anyone wanted two rules. Everything else on this page follows from getting that one choice right.
What This Decibel Calculator Does
It works in three modes. The first converts a ratio between a measured value and a reference value into decibels, applying either the 10-log or the 20-log rule depending on whether the quantities are power-like or amplitude-like. The second reverses that, turning a decibel figure back into the ratio it represents, and reports both interpretations so you can see immediately how much the choice costs you. The third combines several decibel levels into one, which is the operation that cannot be done by adding the numbers.
The results grid stays live in every mode. It shows the power ratio and the amplitude ratio corresponding to the headline decibel figure, the combined level of whatever list you have entered, and how far that combined level sits above the loudest single item in the list. That last figure is the one that surprises people: adding a second source of equal strength gains you three decibels, and adding a source that is ten decibels quieter gains you less than half a decibel.
What this page deliberately does not do is attach a reference. A bare decibel figure is a ratio and nothing more. If your figure carries a suffix — dBm, dBW, dBV, dBA, dB SPL — then a specific reference has been fixed and you are working with an absolute quantity. The dBm to watts converter handles the radio case, and the sound level converter handles acoustic scales.
How to Use It
- Choose the mode that matches your question. Ratio to decibels for gains and losses you have measured, decibels to ratio for reading a specification, and add-levels for combining sources.
- Set the quantity type honestly. Watts, milliwatts, acoustic intensity and optical power are power-like. Volts, amps, sound pressure and any signal amplitude are amplitude-like. If in doubt, ask whether squaring your quantity would give something proportional to power.
- Enter values in matching units. The ratio is dimensionless, so as long as both boxes use the same unit the answer is right. Mixing watts with milliwatts silently introduces a thirty-decibel error.
- Use negative decibels for losses. A cable that passes a third of its input power is at about minus 4.8 dB. There is nothing special about the negative sign; it just means the measured value is below the reference.
- Combine only levels on the same scale. The add-levels mode assumes every figure shares one reference. Combining a dBm figure with a dBA figure produces a number with no physical meaning.
The Formula: 10 Log Versus 20 Log
For power-like quantities the definition is L = 10 × log10(P ÷ Pref). For amplitude-like quantities it is L = 20 × log10(A ÷ Aref). Both are set out in NIST Special Publication 811, the guide for the use of the International System of Units, which also makes the point that the reference quantity must always be stated for a decibel figure to mean anything absolute.
The two rules are not independent conventions; the second follows from the first. Power in a linear system is proportional to the square of amplitude, so a ratio of amplitudes A÷Aref corresponds to a power ratio of (A÷Aref)². Substituting that into the power definition brings the square out of the logarithm as a factor of two, and ten times two is twenty. Both rules therefore give the same number of decibels for the same physical change, which is exactly the property that makes the decibel useful across different quantity types. The decibel is not an SI unit, but it is accepted for use with the SI, and the BIPM SI Brochure covers how logarithmic ratio quantities are handled alongside the SI proper.
Work the default through. A measured power of 100 W against a reference of 1 W is a ratio of one hundred, and ten times the base-ten logarithm of one hundred is 20 dB. The same physical change expressed in voltage would be a voltage ratio of ten, because power goes as the square, and twenty times the logarithm of ten is also 20 dB. Two different rules, two different ratios, one identical answer.
Combining levels uses a third operation. Each level is converted back to a linear power ratio, those ratios are added, and the total is converted back: Ltotal = 10 × log10(Σ10(Li÷10)). Take the default list of 85, 85 and 79 dB. The two 85s combine to 88.01 dB, and adding the 79 brings the total to 88.52 dB. A source six decibels below each of the others contributed about half a decibel to the total.
Why You Cannot Just Add Decibel Numbers
Decibels are logarithms, and logarithms turn multiplication into addition, not addition into addition. Adding two decibel figures is the correct operation when the underlying quantities multiply — a chain of amplifiers and cables, where each stage multiplies the signal by its own gain. In that case a +20 dB amplifier followed by a −3 dB cable genuinely gives +17 dB overall, and the addition is exact.
Adding two decibel figures is wrong when the underlying quantities add — two independent sources arriving at the same point, two machines running in the same room, two noise contributions in a receiver. There the powers add, not the logarithms, and you have to leave the decibel scale, add, and come back. Two 85 dB machines make 88 dB, not 170.
The test is simple. Ask whether the second thing multiplies the first or sits alongside it. Gain, loss, attenuation and amplification multiply, so their decibels add. Sources, contributions and independent signals add, so their decibels must be combined logarithmically. This page separates the two operations into different modes precisely so you have to decide which one you mean.
The combination arithmetic also has a ceiling worth knowing. Any number of identical sources combines to ten times the logarithm of their count above one source. Two give 3 dB, four give 6 dB, ten give 10 dB, and one hundred give 20 dB. Doubling the count always adds the same three decibels, which is why silencing one machine in a room full of them changes so little.
Where the 10-Log and 20-Log Confusion Bites Hardest
The trap is that many quantities can be measured either way, and the same physical situation gets described in both. Sound is the clearest example. Acoustic intensity is power-like and takes 10 log; sound pressure is amplitude-like and takes 20 log. Because sound level meters measure pressure, the 20-log form is what generates the familiar sound pressure level figures, and the reference of twenty micropascals is what makes those figures absolute rather than relative.
Electronics has the same double life. Amplifier gain quoted as a power gain takes 10 log; the same amplifier's voltage gain takes 20 log. Those give identical decibel figures only when the input and output impedances match. When they do not, a stated voltage gain in decibels and a stated power gain in decibels genuinely differ, and neither is wrong — they answer different questions. Reading a datasheet without checking which one is quoted is how designs end up with the wrong headroom.
Optics adds a third variant. Fibre attenuation is quoted in decibels per kilometre against optical power, so it is 10 log. Photodetector responsivity, however, converts optical power to electrical current, so a 3 dB drop in optical power becomes a 6 dB drop in electrical power downstream. That factor-of-two shift catches people out routinely, and it is the same 10-versus-20 issue wearing a different coat.
Reference Points Worth Memorising
Three decibels is a factor of two in power and a factor of about 1.414 in amplitude. Six decibels is a factor of four in power and two in amplitude. Ten decibels is exactly ten in power and about 3.162 in amplitude. Twenty decibels is a hundred in power and ten in amplitude. Those four cover most of what you will ever need to estimate mentally, and this page will confirm any of them in a second.
One decibel is a factor of about 1.26 in power, which is roughly a twenty-six per cent change. That figure is worth carrying because it sets the scale of what a single decibel represents: a change large enough to matter in a link budget, and small enough that a one-decibel measurement error is common. In acoustics a one-decibel change in level is around the threshold of what most listeners can detect at all under controlled conditions.
For the pure logarithm arithmetic underneath all of this, the logarithm calculator and the exponent calculator handle the base-ten operations directly, and the ratio calculator covers proportions before they reach the logarithmic stage. If you need to express a change as a percentage instead, the percentage change calculator does that conversion.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Adding decibel levels arithmetically — two 85 dB sources give 88 dB, not 170. Convert to linear, add, convert back.
- Using the wrong factor — ten for power-like quantities, twenty for amplitude-like ones. Choosing wrongly doubles or halves the answer.
- Treating a bare dB figure as absolute — decibels are a ratio. Without a stated reference such as dBm or dB SPL, the number describes a change and not an amount.
- Combining levels from different scales — an A-weighted acoustic level and a radio power level cannot be summed, however similar their units look.
- Mixing units in the ratio boxes — watts against milliwatts introduces a silent thirty-decibel error, because the tool has no way to know the two boxes disagree.
Related Free Tools From Arb Digital
For absolute radio power on a fixed reference use the dBm to watts converter. Acoustic scales are covered by the sound level converter, and workplace exposure limits by the noise exposure calculator. The underlying maths lives in the logarithm calculator, the exponent calculator and the ratio calculator. Linear power units convert with the power converter, and proportional changes with the percentage change calculator. The full free online tools hub lists everything Arb Digital publishes.
Frequently Asked Questions
Ten times the logarithm applies to power-like quantities such as watts, intensity and optical power. Twenty applies to amplitude-like quantities such as volts, current and sound pressure. The twenty exists only because power is proportional to the square of amplitude.
Convert each level back to a linear ratio, add the ratios, then convert the total back to decibels. Two equal sources always combine to about three decibels above one of them, because doubling the power adds ten times the logarithm of two.
No. A decibel is a logarithmic ratio between two quantities of the same kind, used for radio power, amplifier gain, optical loss and signal-to-noise as much as for acoustics. Sound levels use decibels against a specific pressure reference, which is what makes them absolute.
That the measured quantity is below the reference. Minus three decibels is half the power, minus ten is a tenth, and minus twenty is a hundredth. Attenuation and loss are normally written this way.
Because ten times the base-ten logarithm of two is 3.01. The decibel scale compresses ratios, so a factor of two in power is a small step on it. That compression is the reason the scale is useful when quantities span many orders of magnitude.
Very little. A source ten decibels below another raises the total by about 0.41 decibels, and one twenty decibels below raises it by about 0.04. This is why reducing the loudest contributor is usually the only change that moves a combined level.
This page handles plain decibels, which are ratios with no fixed reference. The dBm converter works with an absolute power scale referenced to one milliwatt, which is why its results come out in watts rather than as a ratio.
This tool is provided for educational and engineering study use. It performs decibel arithmetic on ratios you supply and does not apply weighting curves, measurement standards or exposure limits, so treat its output as arithmetic rather than a compliance measurement.
