Logarithms Without Fear
The word "logarithm" makes many people nervous, but the idea is straightforward: a logarithm is simply a way of asking "what power do I need to raise this base number to, in order to get that result?" Every time you calculate a decibel value you are using a logarithm, and every time you look up an S-meter reading or check a receiver sensitivity spec, the number in front of you came from one. This lesson explains how logarithms work, how to use them confidently, and the three rules that make logarithm arithmetic easy.
What is a logarithm?
A logarithm answers the question: to what power must I raise the base to get this number?
For base 10 (which is the one used in electronics), this is written log10 or just log:
log10(100) = 2 because 102 = 100
log10(10) = 1 because 101 = 10
log10(1) = 0 because 100 = 1
The base-10 logarithm is the one used in electronics unless otherwise stated. When you see "log" in a formula in this course, it always means log base 10.
Logarithms as exponents in reverse
Powers and logarithms are inverse operations — each undoes the other:
- 103 = 1000 → log(1000) = 3
- log(10) = 1 → 101 = 10
Think of them as opposite sides of the same equation. If you know the exponent, raising 10 to that power gives you the result. If you know the result and want the exponent, you take the logarithm.
Key log values to know
Most decibel calculations can be done mentally once you know these values:
| Number | log10(number) | Useful because |
|---|---|---|
| 1 | 0 | log(1) = 0 always, regardless of base |
| 2 | 0.301 | Doubling power = +3 dB (10 × 0.301 ≈ 3) |
| 3 | 0.477 | Tripling power ≈ +4.77 dB |
| 4 | 0.602 | 4× power = +6 dB (0.301 × 2 = 0.602) |
| 5 | 0.699 | 5× power ≈ +7 dB; 1/2 power ≈ −3 dB |
| 8 | 0.903 | 8× power ≈ +9 dB (three doublings) |
| 10 | 1 | 10× power = +10 dB always |
| 100 | 2 | 100× power = +20 dB |
| 1 000 | 3 | 1000× power = +30 dB |
| 1 000 000 | 6 | 1 million × power = +60 dB |
The two most important values to remember are log(2) ≈ 0.3 and log(10) = 1. Almost everything else can be derived from those two.
The logarithmic scale
On a linear scale, equal steps mean equal additions (1, 2, 3, 4, 5…). On a logarithmic scale, equal steps mean equal multiplications. Moving one step up on a log scale multiplies the value by 10:
- 1 → 10 → 100 → 1 000 → 10 000
- 0 → 1 → 2 → 3 → 4 (these are the logarithms)
This is why a logarithmic scale is so useful for quantities that span many orders of magnitude. Power levels in radio systems range from femtowatts (10−15 W) at the receiver input to kilowatts (103 W) at the transmitter output — a range of 1018. On a linear scale, you could not fit both on the same graph. On a log scale (in dBm), that range is −120 dBm to +60 dBm — a span of 180 units that fits comfortably.
A linear scale spaces values by equal additions; a logarithmic scale spaces them by equal multiplications. The same range of 1 to 1,000,000 fits in six steps on the log scale.
View LargerThe three log rules
Three algebraic rules govern logarithms. These are what allow you to convert multiplication into addition — which is why dB values can be added where power ratios would have to be multiplied.
Rule 1: Product rule
Example: log(1000) = log(10 × 100) = log(10) + log(100) = 1 + 2 = 3 ✓
Rule 2: Quotient rule
Example: log(100) = log(1000 / 10) = log(1000) − log(10) = 3 − 1 = 2 ✓
Rule 3: Power rule
Example: log(106) = 6 × log(10) = 6 × 1 = 6 ✓
Example: log(210) = 10 × log(2) = 10 × 0.301 = 3.01 (so 210 = 1024 ≈ 103)
The power rule is especially useful in voltage decibel calculations. The formula dB = 20 × log(V2/V1) comes directly from the power rule: since power is proportional to V2, and log(V2) = 2 × log(V), the factor of 2 moves to the front, turning 10 × log into 20 × log.
The anti-logarithm
The anti-log (or inverse logarithm) reverses the process. If log(x) = y, then x = 10y. On a scientific calculator the anti-log function is usually labelled 10x or INV LOG.
The dB formula gives: dB = 10 × log(ratio), so ratio = 10(dB/10)
ratio = 10(23/10) = 102.3 = antilog(2.3) ≈ 200
So 23 dB of power gain means the output power is approximately 200 times the input power.
Negative logarithms
Log values below 0 correspond to numbers between 0 and 1:
log(0.01) = −2 because 10−2 = 0.01
log(0.5) ≈ −0.301 because 10−0.301 ≈ 0.5
In decibels, a negative value means a loss or a signal level below the reference. A gain of −3 dB means the output is half the input. A signal level of −90 dBm means the power is 10−9 mW = 1 nanowatt.
Natural log vs log base 10
There are two common logarithm bases: base 10 (log10, also written "log") and base e (loge, written "ln" or the "natural logarithm"), where e ≈ 2.718. The natural logarithm appears in physics, mathematics and engineering in places where growth, decay and rate-of-change equations arise. In electronics and radio, it is used in some noise and filter calculations.
However, for all decibel calculations — dB, dBm, dBW, dBi, dBd — you always use log base 10. When an electronics formula says "log" without specifying a base, it means log10. Do not confuse this with the "ln" button on your calculator.
Using logs in electronics calculations
Every decibel formula you will encounter in this course is either:
- dB = 10 × log(P2 / P1) for power ratios
- dB = 20 × log(V2 / V1) for voltage or current ratios
Both are applications of the base-10 logarithm. When you add two dB values, you are using the product rule — you are multiplying the corresponding power ratios. When you subtract, you are using the quotient rule — you are dividing.
EIRP = 50 dBm − 3 dB + 6 dBi = 53 dBm (= 200 W EIRP)
This addition is only valid because we are working in logarithmic units. In watts: 100 W × 0.5 (loss) × 4 (gain) = 200 W — the same result, but requiring multiplication instead of addition.
Mental arithmetic shortcuts
In practice, you do not need to calculate logarithms from scratch. Memorising these two facts gets you through most problems:
- +3 dB ≈ double the power (because log(2) ≈ 0.3, so 10 × 0.3 = 3)
- +10 dB = 10 × the power (because log(10) = 1, so 10 × 1 = 10)
From those two building blocks:
- +6 dB = double twice = 4× power
- +9 dB = double three times = 8× power
- +13 dB = +10 + 3 dB = 10 × 2 = 20× power
- +20 dB = 100× power
- +30 dB = 1000× power
- −3 dB = half the power
- −10 dB = one tenth the power
Combining these steps lets you mentally estimate most conversions that come up in everyday radio work, without touching a calculator.
Frequently Asked Questions
Do I need to understand logarithms to use dB in practice?
For everyday use of dB values — reading specs, calculating link budgets, setting power levels — you just need the key reference points (3 dB = double, 10 dB = ×10). Understanding the underlying logarithm becomes important when you need to convert between dB and actual power or voltage ratios, which comes up regularly when checking transmitter power, receiver sensitivity, or antenna gain specifications.
Why is log(2) approximately 0.3?
Because 100.301 = 2.000 exactly, and 0.301 rounds to 0.3. The approximation is accurate enough for mental arithmetic — 3 dB implies a ratio of 100.3 = 1.995, which is very close to 2. For precise calculations you should use 0.301, but for everyday radio work the 3 dB = ×2 approximation is universally accepted.
What is the logarithm of zero?
The logarithm of zero is undefined — it would require 10y = 0, but no finite value of y achieves that. As a number approaches zero from above, its log approaches negative infinity. In practice, this means you cannot express zero power in dB: you need a positive power level for any dB calculation to make sense.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.