Logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number.
At a glance
Key signals
40% cross fields · reaches 5 more
- Mathematics
- Algebra
- Explanation
- Examples
- Misconception
- Sourced relations 5
- Attribution
Dependencies
What this concept builds on and what it makes possible — derived from the atlas’s dependency, causal and structural relations, not from every related edge.
Enables · leads to
Decibelis derived fromLogarithmEstablished
decibel is derived from the logarithm.
Mechanism: The decibel is defined as ten times the base-ten logarithm of a power ratio.
- Wikipedia (English & German editions) verifiedmoderate evidence
Equal temperamentis derived fromLogarithmEstablished
equal temperament is derived from the logarithm.
Mechanism: Each semitone is the twelfth root of two, an equal ratio on a logarithmic pitch scale.
- Wikipedia (English & German editions) verifiedmoderate evidence
Sound pressure levelis derived fromLogarithmEstablished
sound pressure level is derived from the logarithm.
Mechanism: SPL is twenty times the log of the pressure ratio, matching how loudness is perceived.
- Wikipedia (English & German editions) verifiedmoderate evidence
Structural role & consequence
Interpreted from the current atlas graph — what the connections mean, not just how many there are.
Removing this node lengthens the structural route between Decibel and Sound pressure level from 2 to 4 steps (they stay connected — alternative routes exist).
structural · Structural removal simulation — not a historical or causal counterfactual.
40% of its relationships cross field boundaries, reaching 5 other disciplines — unusual in a discipline where most concepts stay within their field.
structural · Structural graph analysis — not a claim of importance, causation or history.
Currently dark in the atlas: no key date stored · 5 of 5 of its relations lack claim-level evidence.
atlas representation · Describes the current Thinking OS representation, not the state of the world.
Structural neighbourhood: 5 → 14 → 36 concepts reachable within 3 hops.
structural · Structural reach — being reachable is not the same as being understood.
cross-field
3 within-field, 2 cross-field
Strengths & constraints
Strengths
- Cross-disciplinary connector — 40% of its relationships cross field boundaries. structural
Constraints
- Evidence coverage currently thin in the atlas — few of its relationships carry claim-level evidence. atlas representation
- No dated history stored — the atlas records no key date for this concept. atlas representation
Conditions
- Read structurally — most of its relationships carry no external evidence yet, so claims here are graph-derived. structural
Dependency radial
What this concept builds on (left) and what it makes possible (right) — derived from dependency and causal relations.
Seen through each discipline
How this concept sits in each of its fields — derived from its real connections in the graph, not asserted.
Through this lens it connects to Exponentiation, Order of magnitude and Equal temperament.
Check yourself
A quick check against a common misconception. Nothing is scored — picking the tempting-but-wrong answer just flags an idea worth revisiting.
Which statement is correct?
Moving up a log scale by one step adds a fixed amount.
Each step up a log scale multiplies by the base (usually ten), so equal spacings mean equal ratios, not equal differences.
Look for: Learner reads a log axis as if the gaps were linear.
Related ideas to explore
Concepts that look related but are not yet connected here — candidates for a connection to reason about, not established links.
This idea also appears in…
The same structure shows up in other disciplines. These are real recurrences drawn from the graph — a starting point for asking “what carries over, and what changes?”
Scale31 disciplines · 44 concepts
Concepts
shares a mental model · crosses a discipline boundary
shares a mental model · crosses a discipline boundary
shares a mental model
shares a mental model
shares a mental model
shares a mental model
A logarithm answers the question 'how many times do I multiply this number by itself to reach that one?' Since 10×10×10 = 1000, the logarithm (base 10) of 1000 is 3. Logarithms turn big jumps into small steps.
The logarithm is the inverse of exponentiation: log_b(x) is the power to which base b must be raised to give x. Because it converts multiplication into addition, it compresses vast ranges — the basis of decibels, pH and the Richter scale.
Mental models at work here
- Connects 5 other ideas across 2 disciplines.
- A cross-disciplinary bridge — its connections reach into 5 other fields.
- Most of its connections are of the “Dependency” kind.
- It exercises 1 reusable thinking pattern.
Derived from the graph’s real structure — observations, not a score.
Sources
- Wikipedia (English & German editions) verifiedmoderate evidence
- Wikidata verifiedmoderate evidence