Logarithms
The inverse of exponentiation — what logarithms are, how they work, and why they're the right tool for reasoning about scale and growth.
What a Logarithm Is
The logarithm is the inverse of exponentiation. If:
bⁿ = x
then:
log_b(x) = n
Read: “the logarithm base b of x is n.” It asks: what power do I raise b to in order to get x?
log₂(8) = 3 because 2³ = 8
log₁₀(1000) = 3 because 10³ = 1000
log₁₀(0.01) = −2 because 10⁻² = 0.01
The exponent is the answer. The logarithm is the question.
The Three Bases That Matter
Base 10 (common logarithm, log) — the natural choice for order-of-magnitude reasoning. log₁₀(x) tells you roughly how many digits x has.
log(1) = 0
log(10) = 1
log(100) = 2
log(1000) = 3
log(1,000,000) = 6
Base e (natural logarithm, ln) — e ≈ 2.718. The natural base for continuous growth and calculus. ln(x) appears in all growth/decay problems, probability distributions, and calculus.
Base 2 (binary logarithm, log₂) — the natural base in computer science. log₂(n) = number of bits needed to represent n values, number of times you can halve n before reaching 1.
log₂(8) = 3 → 8 values need 3 bits
log₂(1024) = 10 → binary search on 1024 items takes 10 steps
Laws of Logarithms
These correspond directly to the laws of exponents:
| Law | Formula | Example |
|---|---|---|
| Product | log(ab) = log(a) + log(b) | log(200) = log(2) + log(100) = 0.3 + 2 = 2.3 |
| Quotient | log(a/b) = log(a) − log(b) | log(50) = log(100) − log(2) = 2 − 0.3 = 1.7 |
| Power | log(aⁿ) = n·log(a) | log(8) = 3·log(2) = 3 × 0.3 = 0.9 |
| Base change | log_b(x) = log(x)/log(b) | log₂(x) = log(x)/log(2) = log(x)/0.3 |
Why product → sum: log(ab) = log(a) + log(b) because exponents add on multiplication. Logarithms turn multiplication into addition — this is exactly why slide rules worked, and why log tables were used for centuries of scientific calculation.
Key Values to Know
Base 10
log(2) ≈ 0.301
log(3) ≈ 0.477
log(5) ≈ 0.699 (= 1 − log(2), since log(2×5) = log(10) = 1)
log(7) ≈ 0.845
log(e) ≈ 0.434
From these, you can derive most others using the laws:
log(4) = 2·log(2) ≈ 0.602
log(6) = log(2) + log(3) ≈ 0.778
log(8) = 3·log(2) ≈ 0.903
log(9) = 2·log(3) ≈ 0.954
Natural log
ln(1) = 0
ln(e) = 1
ln(2) ≈ 0.693
ln(10) ≈ 2.303
Conversion: ln(x) = log(x) × ln(10) ≈ log(x) × 2.303
Logarithmic Scale
A logarithmic scale spaces values by their order of magnitude rather than their absolute difference. Each equal step on the scale represents a multiplication, not an addition.
When to use it: whenever the data spans many orders of magnitude. Earthquake magnitudes (Richter scale), sound intensity (decibels), star brightness (magnitude), pH, frequency in music, stock returns over decades.
The Richter scale: magnitude 7 is 10× more powerful than magnitude 6, not just one unit more powerful. A magnitude 9 earthquake releases 1000× the energy of a magnitude 7.
Decibels: 10 dB difference = 10× more intense. 20 dB = 100×. 60 dB difference = 10⁶× more intense.
In Fermi estimation: mentally working on a log scale lets you reason about whether an answer is 10³ or 10⁶ before worrying about the coefficient. The exponent is usually the important thing.
Logarithms and Growth
Exponential growth (output grows as a power of input) becomes linear on a log scale. If you plot y = 10ˣ on a log scale for y, you get a straight line.
This is why log-log plots and semi-log plots are used in science and engineering — they linearise power-law and exponential relationships, making trends easy to see and quantify.
Doubling time: a quantity growing at rate r per period doubles when:
(1 + r)ⁿ = 2
n = ln(2)/ln(1+r) ≈ ln(2)/r ≈ 0.693/r
The rule of 72: n ≈ 72/r% approximates this (72 ≈ 100 × ln(2) × adjustment for discrete compounding).
log vs ln in Practice
In pure mathematics and physics, ln is standard. In common-use calculation and Fermi estimation, log₁₀ is more intuitive because it directly gives order of magnitude.
log₁₀(world population) = log₁₀(8 × 10⁹) ≈ 9.9
log₁₀(distance to Moon in km) = log₁₀(384,000) ≈ 5.6
log₁₀(atoms in human body) ≈ 28
The exponent is the scale. That’s the number worth knowing.
The Slide Rule Intuition
Before calculators, scientists multiplied using slide rules — physical logarithm tables you could slide against each other. To multiply 3 × 7: find log(3) on one scale, log(7) on the other, add them (mechanically, by sliding), read off the antilog. The answer is log⁻¹(log(3) + log(7)) = log⁻¹(0.477 + 0.845) = log⁻¹(1.322) ≈ 21.
Logarithms turned multiplication into addition. That was the entire value — and it was enough to do most of the engineering and science of the 18th and 19th centuries.