What a logarithm does
log₁₀(x) asks: 10 to what power gives x?
log₁₀(10) = 1 log₁₀(100) = 2 log₁₀(1,000,000) = 6 log₁₀(0.001) = −3
Multiplying by 10 adds 1 to the log. Multiplying by a million adds 6. That's the key property:
log(a × b) = log a + log b
So a scale based on logs compresses an enormous range into small, manageable numbers, and turns "ten times bigger" into "one step up". The natural log, ln, works the same way with e ≈ 2.718 instead of 10, and is the right choice whenever something grows or shrinks at a rate proportional to its size.
Decibels: sound and signals
The quietest sound a healthy ear can hear and the loudness of a jet engine nearby differ in intensity by a factor of around a trillion (10¹²). Decibels squash that into 0 to 120:
Level in dB = 10 × log₁₀( P / P₀ )
where P is the power (or intensity) and P₀ a reference. Every +10 dB is 10 times the power. A few consequences that surprise people:
- Doubling the power adds only 3 dB. 10 × log₁₀(2) = 3.01. Two identical machines each at 80 dB make 83 dB together, not 160.
- +20 dB is 100 times the power, and +30 dB is 1,000 times. Going from 60 dB (conversation) to 90 dB (a loud workshop) is a thousandfold increase in sound intensity.
- For voltages and sound pressure, use 20 × log₁₀. Power goes as the square of voltage or pressure, and the 2 from the square comes out in front (log x² = 2 log x). Doubling a voltage is +6.02 dB.
An amplifier delivers 50 W; the reference is 1 mW. Its output level is
10 × log(50 ÷ 0.001) = 10 × log(50,000) = 10 × 4.699 = 46.99 dBm
The "m" in dBm means the reference is one milliwatt, which is standard in radio and audio engineering.
Type it into the scientific calculator.
Open in the calculatorpH: how acidic is it?
The concentration of hydrogen ions in everyday liquids ranges from about 1 mol/L (strong acid) to 10⁻¹⁴ mol/L (strong alkali). pH turns that into 0 to 14:
pH = −log₁₀ [H⁺]
The minus sign just makes the numbers positive. Pure water at 25 °C has [H⁺] = 10⁻⁷, so pH 7.
- One pH unit is a factor of 10. Lemon juice at about pH 2 has roughly 1,000 times more hydrogen ions than black coffee at about pH 5.
- Lower pH = more acidic: more H⁺ makes log[H⁺] larger (closer to zero), and the minus sign turns that into a smaller pH.
A sample has [H⁺] = 3.2 × 10⁻⁴ mol/L. Its pH is
pH = −log(3.2E-4) = 3.49
Going the other way, [H⁺] = 10−pH, so pH 3.49 gives 10^(−3.49) = 3.2 × 10⁻⁴ mol/L.
Earthquake magnitudes
The original Richter scale (1935) defined magnitude from the logarithm of the largest wave amplitude on a seismograph, so each whole number up meant 10 times the ground motion. Seismologists now use the moment magnitude scale, which was designed to match Richter's numbers for moderate quakes but is based on the energy released. The key rule of thumb:
- +1 magnitude ≈ 10 times the shaking amplitude and about 32 times the energy (101.5 = 31.6).
- +2 magnitude ≈ 1,000 times the energy. A magnitude 7 earthquake releases about a thousand times more energy than a magnitude 5.
That's why a magnitude 6 is a serious event while a magnitude 4 is often barely noticed, even though the numbers look close.
Half-lives and natural logs
Radioactive decay, drug levels in the blood and the discharge of a capacitor all follow the same rule: in each fixed interval, the same fraction disappears. After time t, with half-life T,
N / N₀ = (½)^(t / T)
To find the time, you need to get t out of the exponent, and that's what logs are for. Taking ln of both sides:
t = T × ln(N / N₀) / ln(½)
Carbon-14 has a half-life of 5,730 years. A sample has 20% of the carbon-14 a living organism would have. How old is it?
t = 5730 × ln(0.2) / ln(0.5) = 5730 × (−1.6094) / (−0.6931) ≈ 13,300 years
Sanity check: 25% would be exactly two half-lives (11,460 years), and 20% is a bit less than that, so a bit older. ✓
Doubling time and the rule of 72
The same trick works for growth. Something growing at 5% a year doubles after
t = ln(2) / ln(1.05) = 14.2 years
The "rule of 72" is a mental shortcut: divide 72 by the percentage rate. 72 ÷ 5 = 14.4 years, close enough for most purposes. It works because ln 2 ≈ 0.693 and ln(1 + r) ≈ r for small r, with 72 chosen instead of 69.3 because it divides neatly by 2, 3, 4, 6, 8, 9 and 12.
The log laws you'll actually use
| Rule | Example |
|---|---|
| log(ab) = log a + log b | Two gains of 10 dB and 6 dB make 16 dB in total |
| log(a/b) = log a − log b | A signal at 46 dBm through a cable losing 3 dB arrives at 43 dBm |
| log(aᵏ) = k log a | Why voltage decibels use 20 log instead of 10 log |
| log_b(x) = ln x / ln b | log₂(1024) = ln(1024) ÷ ln(2) = 10 |
Logs are only defined for positive numbers. If a calculation asks for the log of zero or a negative number, the calculator will refuse, and that usually means a sign error earlier on.