10³ is a thousand: you know that already. But what about 10⁻³? And that mol⁻¹ stuck onto a number? At school, powers almost always go up. In science they go down too, and that’s where logarithms come from.
10⁻³ Negative exponent: divide
- Say it
- 10⁻³ is read “ten to the minus three”.
- It means
- A negative exponent means divide: 10⁻³ = 1 / 10³ = 1/1000 = 0.001.
If 10³ is multiplying 1 by 10 three times, 10⁻³ is dividing 1 by 10 three times. That’s how you write tiny numbers without filling the page with zeros: 3×10⁻⁴ is 0.0003.
Real example · in the lesson Why does the computer say 0.1 + 0.2 isn't 0.3?
|δ| ≤ u = 2⁻⁵³: the rounding error is never more than 1 / 2⁵³, roughly one ten-quadrillionth.
J/(mol·K) Compound units: mol⁻¹ and J/(mol·K)
- Say it
- J/(mol·K) is read “joules per mole per kelvin”. mol⁻¹ is read “per mole”.
- It means
- A unit with a slash or a −1 exponent says “each”: kJ/mol is kilojoules per mole. mol⁻¹ is the same as /mol.
Units multiply and divide just like numbers. J/(mol·K) is joules for every mole and every kelvin: the dot inside means “and”, and the brackets group what goes underneath. And since dividing is multiplying by the power −1, mol⁻¹ means “per mole”.
Real example · in the lesson What is a wall made of? Atoms and elements without fear
N_A = 6.022 140 76×10²³ mol⁻¹: that many particles in every mole.
Real example · in the lesson Why doesn't diamond turn into graphite?
R = 8.314 J/(mol·K): the gas constant.
h⁽ˡ⁾ Superscript in brackets: not a power
- Say it
- h⁽ˡ⁾ is read “h of layer l”.
- It means
- A number or letter up top and in brackets is a label, like a subscript that moved upstairs so it doesn't clash with the subscripts already there.
Neural networks have layers, and each layer has its numbers. h⁽¹⁾ are the first layer’s and h⁽²⁾ the second’s. The brackets warn you that the number up top isn’t raising anything to a power.
Real example · in the lesson Logistic regression or neural network? Same neuron
h⁽ˡ⁾ = σ(z⁽ˡ⁾): what comes out of layer l.
e The number e and the exponential
- Say it
- “e”, Euler's number. e^(−t/τ) is read “e to the minus t over tau”. exp(x) is the same as eˣ.
- It means
- e ≈ 2.718 is a number, like π. Raised to a power, eˣ describes things that grow or die away at a rate proportional to how much there is.
It shows up when something changes in proportion to itself: interest that’s reinvested, a cup that cools faster the hotter it is, or what share of molecules have enough energy to react. When the exponent is very long, it’s written exp(…) to make it easier to read.
Real example · in the lesson Why doesn't diamond turn into graphite?
k = A·e^(−Eₐ/RT): the higher the temperature, the more collisions arrive with energy to spare.
log The logarithm: how many times do I multiply?
- Say it
- log₁₀ 1000 is read “log base ten of a thousand”. ln is the logarithm in base e.
- It means
- The logarithm is the power question backwards: what do I raise the base to so that this number comes out? log₁₀ 1000 = 3, because 10³ = 1000.
For very big or very small numbers, the logarithm counts the zeros: log₁₀ 1,000,000 = 6 and log₁₀ 0.001 = −3. That’s why it turns multiplications into additions and lets you fit things of wildly different sizes on one chart. log₂ does the same in base 2, the computers’ base, and ln does it in base e.
Real example · in the lesson Why doesn't diamond turn into graphite?
ln k = ln A − (Eₐ/R)(1/T): with logarithms, the Arrhenius curve becomes a straight line.
⌊x⌋ The floor: ⌊x⌋
- Say it
- ⌊x⌋ is read “floor of x”.
- It means
- Brackets with only the bottom foot round down: ⌊3.7⌋ = 3 and ⌊−1.2⌋ = −2.
It keeps the whole number just below. Combined with the logarithm, it gives how many digits a number has before the decimal point, minus one.
Real example · in the lesson How much is 7×10²⁷? Huge numbers without fear
n = ⌊log₁₀|x|⌋: the exponent in scientific notation is the floor of the logarithm.
Try it
Move the exponent from −9 to 9. Watch how the decimal, the fraction and the prefix change, and notice that the base-10 logarithm is exactly the exponent.
Challenge
Sources
- Powers, e, exp, log, ln and the floor: ISO 80000-2:2019, Quantities and units — Part 2: Mathematics (iso.org).
- How to write compound units and the SI prefixes (table 7): BIPM, The International System of Units (SI Brochure), 9th ed., 2019 (bipm.org).
How this cheat sheet was made: written with Claude’s help from the sources above and translated from the Spanish original. The examples are copied from the site’s lessons. The drawings are original.