Common Logarithm
Understanding the Common Logarithm
A logarithm answers the question, to what power must the base be raised to produce this number. The common logarithm fixes that base at 10, so log10(100) is 2, log10(1) is 0, and log10(0.001) is minus 3. The defining property that makes logarithms so useful is that they convert multiplication into addition: the log of a product equals the sum of the logs. Chains of RF components multiply their gain and loss ratios together, so converting each ratio to its logarithm lets an engineer simply add and subtract numbers instead of multiplying a long string of factors.
RF signals demand this compression. A transmitter may radiate a watt while the receiver listens for a femtowatt, a span of fifteen orders of magnitude. Writing those numbers linearly is unwieldy and error prone. On a base-10 logarithmic scale the same span is just 150 units, and the decibel formalizes it. Because each factor of ten in power equals exactly 10 dB, an engineer reads orders of magnitude directly off the number: 30 dB is a thousandfold, 60 dB is a millionfold.
From Logarithm to Decibel to dBm
The decibel is defined directly from the common logarithm of a ratio. For power, one decibel is one tenth of a bel, and a bel is the base-10 logarithm of a power ratio, so the working formula is 10 times log10 of the power ratio. Voltage and current relate to power through a square law in a fixed impedance, which introduces the factor of 20 for amplitude ratios. Absolute levels follow by fixing the reference: dBm references one milliwatt and dBW references one watt, so a level in dBm is 10 times log10 of the power expressed in milliwatts. The reverse operation, the antilogarithm, raises 10 to the appropriate power to recover the linear value, and fluency in both directions is essential for link budgets, noise figure, and intermodulation math.
Common Logarithm and Decibel Equations
y = log10(x) ⇔ x = 10y
Power and voltage decibels:
dB = 10 log10(P2/P1) = 20 log10(V2/V1)
Absolute level and antilog:
P(dBm) = 10 log10(P / 1 mW), P(mW) = 10P(dBm)/10
Where x = a positive number, y = its common logarithm, P = power, V = voltage. Example: a power ratio of 2 is 10 log10(2) ≈ 3.01 dB, and 0 dBm equals exactly 1 mW.
Logarithm Bases and Decibel Reference Points
| Quantity | Base / reference | Relation | RF use |
|---|---|---|---|
| Common log | Base 10 | log10(x) | Decibels, link budgets |
| Natural log | Base e | ln(x) = 2.3026 log10(x) | Decay, neper, math |
| Power ratio 2 | 1 mW (for dBm) | ≈ 3.01 dB | Doubling power |
| Power ratio 10 | 1 W (for dBW) | = 10 dB | One decade |
| Voltage ratio 2 | Fixed impedance | ≈ 6.02 dB | Doubling amplitude |
Frequently Asked Questions
What is the common logarithm?
It is the logarithm to base 10: the common logarithm of x is the exponent to which 10 must be raised to give x, so log10(1000) = 3. Logarithms turn multiplication into addition, which suits RF engineering where signals span many orders of magnitude. The base-10 choice maps cleanly onto decibels, where each factor of ten in power is exactly 10 dB.
How does it define the decibel?
A decibel is 10 times the common logarithm of a power ratio: dB = 10 log10(P2/P1). Since power is proportional to voltage squared in a fixed impedance, voltage ratios use 20 instead. This lets engineers add gains and subtract losses along a chain. A power ratio of 2 is about 3 dB, 10 is 10 dB, and 1000 is 30 dB.
What is the antilogarithm in RF?
It is the inverse of the common logarithm: the antilog of y is 10 to the power y. It converts decibels back to a linear ratio or level. To get milliwatts from dBm, compute 10 to the dBm/10. So 30 dBm is 1000 mW (1 W) and minus 90 dBm is one picowatt. Moving between the dB and linear domains is core to link-budget and noise math.