Intuition The one core ideaSome real-world quantities span a ludicrous range — from a whisper to a jet, sound intensity multiplies a trillion-fold. A logarithmic scale is a ruler that counts how many times you multiplied , so gigantic multiplicative jumps become small, evenly-spaced additive steps.
This page assumes nothing . Before you can read the parent topic , you must own every symbol it fires at you. We build them one at a time, each on top of the last, each anchored to a picture.
Intuition How to read the coloured boxes on this pageThroughout the vault, boxed labels flag what kind of statement you're reading, so you can skim or study deliberately: [!definition] = a term is being pinned down, [!formula] = a key equation and its derivation, [!intuition] = a plain-words "why," [!example] = a worked case, [!mistake] = a trap and its fix. Words in highlight are the terms worth memorising; a line written Question ::: Answer is a hidden self-test — cover the answer and try first.
Definition Exponent notation1 0 n counter of how many times you multiply 10 by itself .
1 0 1 = 10 , 1 0 2 = 10 × 10 = 100 , 1 0 3 = 1000 , … exponent or power .
WHAT the picture shows: think of a number line where each step is not "+1 apple" but "×10." One step lands on 10, two steps on 100, three on 1000. The distances jump enormously but the number of steps grows gently: 1, 2, 3.
WHY the topic needs this: decibels, Richter and pH all live in the world of 1 0 something
Intuition Negative and fractional exponents
1 0 − 3 = 1 0 3 1 = 1000 1 = 0.001 negative exponent means "divide instead of multiply" — it takes you to tiny numbers. This is exactly the world of [ H + ] 1 0 0.5 = 10 ≈ 3.162 fractional exponent means "part-way between two whole powers." Half a step from 1 0 0 = 1 1 0 1 = 10 3.162 geometric middle (see Orders of Magnitude ).
The exponent turns a count of multiplications into a big number . The logarithm runs that arrow backwards .
log 10 x log 10 x
"10 to what power gives me x
The answer is the logarithm. Symbolically:
log 10 x = n ⟺ 1 0 n = x . 10 base — the number we are repeatedly multiplying.
x positive
log 10 x x > 0 Why? No power of 10 ever lands on zero or a negative number: 1 0 n x has no answer when x ≤ 0
log 10 0 undefined — you'd have to go infinitely far down the ×10 ladder (1 0 − 1 , 1 0 − 2 , … log 10 ( negative ) undefined — the ×10 ladder never crosses to the other side of zero.
Why the topic needs this: every scale feeds log 10 strictly positive , which is exactly why they're legal inputs. A silence of literally zero intensity would push the decibel formula to − ∞
WHAT it looks like: in the figure below, the top arrow (1 0 n up from a count to a big number; the bottom arrow (log 10 back down from the big number to the count. They undo each other.
Worked example Reading logs off the picture
log 10 1000 = 3 1 0 3 = 1000 log 10 10 = 1 log 10 1 = 0 log 10 0.001 = − 3 1 0 − 3 = 0.001 Small (but still positive) number → negative log. Hold onto this: it is the entire reason pH carries a minus sign.
log Why it feels right: bigger input usually means bigger output.
The fix: the log is the exponent , not the number. log 10 trillion (1 0 12 12
WHY the topic needs it: every scale in the parent — L = 10 log 10 ( ⋅ ) M = log 10 ( ⋅ ) pH = − log 10 ( ⋅ ) log 10 e Change of Base Formula for why any base could work.
This is the single fact that makes log scales worth inventing .
WHY it is true, from the picture: log a = 1 0 p p b = 1 0 q q ab = 1 0 p × 1 0 q = 1 0 p + q walked p q , landing at step p + q adding .
Intuition Why this all matters for the whole topicReal systems (ears, earthquakes, acids) grow by multiplying : ×10, ×10, ×10. That's clumsy to plot. The product and quotient rules let a log convert those multiplications and divisions into even, evenly-spaced additions and subtractions : +1, +1, −1. That's what a logarithmic scale is — see Laws of Logarithms for the full family of rules.
1 0 L L = log 10 ( something ) recover that something you raise 10 to the power L something = 1 0 L . 1 0 L always positive — matching the domain rule that only positive numbers can come out of "un-logging."
WHY the topic needs it: the parent constantly runs backwards — "a solution has pH 8.5, find the concentration" needs 1 0 − 8.5 Exponential Functions .
Every scale compares a quantity to a fixed reference , then logs the ratio.
Definition A generic quantityC C 0 all three scales at once without picking a specific physical unit, we use two neutral placeholder letters:
C quantity you measured — always positive, so it's a legal log input.C 0 anchor value (the "zero point" of the scale), same units as C
C C 0 stand-ins . Each real scale swaps in its own letters and numbers:
Scale
quantity C
reference C 0
Decibel
intensity I
I 0 = 1 0 − 12 W/m 2
Richter
amplitude A
reference amplitude A 0
pH
concentration [ H + ]
1 mol/L
Read this carefully: C not one physical thing — in decibels it is intensity, in pH it is hydrogen-ion concentration. They are different physical quantities that happen to share the same mathematical shape , which is the whole reason one formula covers all three.
C / C 0 C / C 0 how many times bigger your quantity is than the anchor. Because both have the same units, those units cancel , leaving a pure number — which is exactly what log
For pH the anchor is 1 mol/L [ H + ] /1 mol/L − log 10 [ H + ]
Intuition Why divide first?log C 0 then the log counts the powers of ten in that answer. If the ratio is below 1 (quantity smaller than the anchor), the quotient rule of Section 2 tells you the log — and hence the scale reading — goes negative.
k L = k log 10 ( C 0 C ) . k
k = 10 decibels (a bel was too coarse, so tenths).k = 1 Richter .k = − 1 pH (the minus flips tiny numbers into friendly positive ones).
That is the only difference between the three famous scales.
WHAT it looks like: same log curve, three different vertical scalings — one stretched ×10, one left alone, one mirrored. Nothing new mathematically.
Definition The multiplierr r how many times bigger the new one is :
r = C old C new . r = 2 r = 10 r = 10 1 r
The picture below shows the dependency order as boxes and arrows: powers of ten and negative exponents feed the logarithm; the logarithm splits into the product/quotient rules (which give the change law) and its exponential inverse; those, with the reference ratio and the step constant, assemble the general scale that becomes decibels, Richter and pH. Read it bottom-up when revising: to trust any box, make sure every box feeding an arrow into it already feels obvious.
Test yourself — each should feel obvious before you open the parent note.
What does the exponent n 1 0 n How many times you multiply 10 by itself.
What is 1 0 − 3 1000 1 = 0.001
What question does log 10 x "10 to what power gives x
For which inputs is log 10 x Only x > 0 log 10 0 log 10 ( negative )
Why is log 10 Because a trillion is 1 0 12
Why is the log of a number between 0 and 1 negative? It's a positive but small number like 1 0 − 3
State the product rule and why it's true. log 10 ( ab ) = log 10 a + log 10 b adds their exponents.
State the quotient rule and why it's true. log 10 ( a / b ) = log 10 a − log 10 b subtracts their exponents.
Why does 1 0 l o g 10 x = x Let n = log 10 x 1 0 n = x 1 0 l o g 10 x = x
Why does log 10 ( 1 0 L ) = L "10 to what power gives 1 0 L L
Why divide by C 0 To get a pure unitless positive ratio "how many anchors' worth," which is all log
What does C A generic placeholder quantity — intensity for dB, amplitude for Richter, [ H + ]
What is C 0 1 mol/L [ H + ]
What does the constant k Sets one step's size (10 = dB, 1 = Richter, −1 = pH).
What is r Δ L = k log 10 r r = C new / C old r
Parent topic — where these foundations get used.Laws of Logarithms — product and quotient rules, formalised.Exponential Functions — the "1 0 L Change of Base Formula — why base-10 is a choice , not a necessity.Orders of Magnitude — "counting powers of ten" as a habit.Weber–Fechner Law — why perception is multiplicative in the first place. 
