Logarithmic Scale Calculator
Logarithmic scale calculator — convert ratios to decibels (power/amplitude), Richter-like magnitude, or pH-style readings. Free online log-scale calculator with visuals and practice.
Ears, seismometers, and pH meters all compress reality with logs — translate any ratio into their dialects.
- Grade level
- 8–12 + College
- Curriculum band
- Difficulty
- Intermediate
- For this topic
- Time to master
- 2 weeks
- With daily practice
- Tools on page
- 8 on page
- Calculator + practice
Calculator
Scale Inputs — ratio + scale
Log Scale Summary
Breakdown
| Property | Value |
|---|---|
| log₁₀(ratio) | 5 |
| dB power equivalent | 50 |
| pH-style equivalent | -5 |
Results
Decibels (power)
50dB
10·log₁₀(ratio) applied
5
Raw tens count underneath
100000×
Input ratio vs reference
10·log₁₀(ratio)
Scale definition
Results apply standard log-scale definitions for the inputs given. The ratio must be strictly positive.
Statistics
Live readouts from your current inputs — all four scales recompute together.
Decibels (power)
50 dB
10·log₁₀(ratio)
log₁₀(ratio)
5
Raw tens count
Ratio
100000×
vs reference
Visualizations
All four scales for this ratio
Scale reading vs ratio (log axis)
Reading intensity
The log curve every scale here rides on
Formula
Four dialects of one idea: multiply the tens-count by the scale factor — or negate it for flipped scales.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| ratio | Ratio | × | Measured over reference; > 0 |
| dB | Decibels | dB | 10× (power) or 20× (amplitude) |
| M | Magnitude | M | Richter-like log₁₀ reading |
| pH | pH-style | pH | Negated log₁₀ reading |
Explanation
Human senses and disaster energies span trillion-fold ranges — logs compress them into numbers people can say out loud.
Derivation
Level = k·log₁₀(ratio): k = 10 (power bel→decibel), 20 (amplitude² law), 1 (magnitude), −1 (flipped pH).
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Form the ratio against the reference.
Take log₁₀ to count the tens.
Apply the scale multiplier.
Interpret against known landmarks.
Confirm the domain held.
State the final reading.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Sound power ratio 100,000.
Inputs
Calculation
10·log₁₀(100000) = 10 × 5 = 50
Result
50 dB
Interpretation
Normal conversation level.
Understanding Your Result
Your Result: Decibels (power): 50 dB
Typical range: Whisper ~20 dB · conversation ~60 dB · jet ~140 dB; quakes M0–M9; pH 0–14.
What This Means
The result is your ratio translated into the chosen scale language, with raw log₁₀ underneath.
Why It Matters
Volume knobs, quake news, and pool kits all speak these scales — converting ratios is reading the news.
Professional Interpretation
Always state power vs amplitude for dB — the 2× factor traps even engineers.
Recommendations
- •State power vs amplitude explicitly
- •Anchor with known landmarks
- •Negate carefully on flipped scales
Learn — The Complete Guide
Log scales compress colossal ratios into speakable numbers. This calculator translates any ratio into four dialects.
The core concept
Reading = k·log₁₀(ratio): the multiplier k sets the language — 10, 20, 1, or −1.
- Scales compress trillion-fold ranges
- Multiplier sets the language: 10/20/1/−1
- Ratios multiply ⟺ readings add
Derivations — why the formulas work
20× — from the square law
- 1P ∝ A²
- 210·log₁₀(A²)
- 3= 10·2·log₁₀(A)
- 4= 20·log₁₀(A) ✓
Stacking — products become sums
- 1dB_total = 10·log₁₀(r₁·r₂)
- 2= 10·log₁₀(r₁) + 10·log₁₀(r₂)
- 3= dB₁ + dB₂ ✓
- 4Gains add; ratios multiply
Where you meet this in real life
Learn this first — prerequisites in order
- 1Common logs: evaluate log₁₀ fluently
- 2Ratios: form measured-over-reference
- 3Orders of magnitude: feel ×10 steps
Careers that use this daily
- •Audio engineer — gain structure in dB
- •Seismologist — magnitude catalogs
- •Lab technician — pH and dilutions
- •RF engineer — link budgets
Why it helps
- •Four scales from one ratio
- •Side-by-side comparison bars
- •Landmark-anchored interpretation
Best practices
- •Confirm power vs amplitude first
- •Anchor with known landmarks
- •Keep the reference stated
Limitations — what this tool does not do
- •Ratio must be strictly positive
- •Simplified Richter (real formula adds calibration)
- •Display rounds to 12 significant digits
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| Middle / Algebra I | 7–9 | HSA-SSE, RP | Read dB values · Ratios · Magnitude sense |
| Physics / Chemistry | 10–12 | HS-PS2, HS-PS1 | dB power vs amp · Richter energy · pH scale |
Timeline — how this idea grew up
1909
pH defined
Sørensen’s flipped log gives chemistry its 0–14 language.
1920s
Decibel adopted
Bell Labs compresses power ratios into telephone-friendly numbers.
1935
Richter scale debuts
Seismology compresses quake energy into single digits.
A little history
- • The bel honored Alexander Graham Bell; the decibel made it practical.
- • Sørensen defined pH to tame tiny concentrations in brewing chemistry.
Exam-day checklist
- • Write the formula with its multiplier.
- • Anchor with landmarks.
- • Check the sign on flipped scales.
- • State power vs amplitude.
Myths vs facts
Practical Tips
- •Confirm power vs amplitude first
- •Anchor: 50 dB chat, M5 moderate
- •Below 1 means negative readings
- •State the reference always
- •Flip sign only for pH-style
Common Mistakes
Assumptions
This calculator makes the following assumptions to provide estimates:
Limitations
Be aware of these limitations when using this calculator:
Frequently Asked Questions
Answers to common questions about this calculator.
How do decibels use logarithms?
Why 10× for power but 20× for amplitude?
How does the Richter scale use logs?
How does pH use logs?
Can the ratio be zero or negative?
How do I use this logarithmic scale calculator?
Related Calculators
Explore more calculators related to this topic.
Related Guides
Deepen your understanding with these educational resources.
Glossary
Technical terms used in mortgage calculations explained in plain language.
- Decibel
- 10·log₁₀ (power) or 20·log₁₀ (amplitude).
- Magnitude
- log₁₀-style quake reading.
- pH-style
- Negated log₁₀ reading.
- Reference
- The denominator the ratio is measured against.
- Landmark
- Known reading (50 dB chat, M5 quake).
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
Diagrams & Illustrations
Visual intuition for every core idea. Select a card to enlarge it.
Did You Know?
Short intuition builders — great for class warm-ups.
Your ears are logarithmic
+10 dB sounds roughly twice as loud but carries 10× the power — Weber–Fechner in action.
M6 vs M5 is 32× energy
One Richter step means ~31.6× more energy — small numbers, terrifying ratios.
pH 2 vs 7 is 100,000×
Lemon juice vs pure water: five pH units = 10⁵ concentration ratio — this page’s default.
The bel was too big
Deci-bel won because whole bels were too coarse — engineers needed the tenth.
Compare Side by Side
One ratio, every language — the same tens-count, four different multipliers.
| Scale | ratio = 100000× | Formula | Reading |
|---|---|---|---|
| dB power | 10·log₁₀ | 10 × 5 | 50 dB |
| dB amplitude | 20·log₁₀ | 20 × 5 | 100 dB |
| Richter-like / pH | ±log₁₀ | ±5 | 5 M / -5 pH |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
Decibels (power): 100000 → 50 dB
Current inputs — live
Study Plan
A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.
Prerequisites
- ▸Common logs (log₁₀ fluency)
- ▸Ratios and proportions
- ▸Orders of magnitude
Mastery progress — 0%
- I can convert any ratio to dB
- I can explain 10× vs 20×
- I can read Richter and pH
- I can stack gains by adding
- 1
Week 1 · ~20 min/day
Read the scales
Goals
- Convert ratios to dB both ways
- Read Richter magnitudes
- Interpret pH values
Practice
Convert 10 ratios into all four readings by hand.
- 2
Week 2 · ~25 min/day
Power vs amplitude
Goals
- Explain the 10× vs 20× split
- Stack dB gains by addition
- Negate correctly for pH-style
Practice
Stack 5 gain chains and verify with multiplied ratios.
- 3
Week 3 · ~20 min/day
Scales in the wild
Goals
- Read quake news quantitatively
- Set audio gain deliberately
- Interpret lab pH slips
Practice
Translate one real headline into ratios and back.
Exam tips
- •Write the formula with its multiplier.
- •Anchor with landmarks (50 dB, M5).
- •Check the sign on flipped scales.
Practice Problems
0/5 solvedTry each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.
Convert ratio 100,000 to dB power.
Convert ratio 100,000 to Richter-like.
Convert ratio 100 to dB amplitude.
Two +10 dB gains in series = ?
A quake releases 1000× the energy of M4. Magnitude?
Printable Worksheet
Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.
- 1.18 × 47 =
- 2.8 ÷ 11 =
- 3.38 × 26 =
- 4.35 ÷ 4 =
- 5.21 − 16 =
- 6.2 + 26 =
- 7.14 ÷ 12 =
- 8.23 ÷ 3 =
- 9.49 ÷ 7 =
- 10.22 − 17 =
- 11.41 + 18 =
- 12.47 − 35 =
Formula Cheat Sheet
Every formula on this page in one printable table.
| Formula | Meaning | Example |
|---|---|---|
| dB = 10·log₁₀(P ratio) | Power decibels | 10⁵→50 dB |
| dB = 20·log₁₀(A ratio) | Amplitude decibels | 10⁵→100 dB |
| M = log₁₀(ratio) | Richter-like | 10⁵→M5.0 |
| pH = −log₁₀[H⁺] | Flipped scale | 10⁻³→pH 3 |
| Stacked dB add | Ratios multiply | +10+10=+20 dB |
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Page Information
2026-09-29
3.1.0-math
Free to use for educational and professional purposes.