pH Logarithm Calculator
pH calculator — pH = −log₁₀[H⁺] with acidic/neutral/basic labeling and [OH⁻] companion. Free online pH logarithm calculator with visuals, practice and study plan.
A flipped logarithm turns hundredths of molarity into a single digit — each step is a tenfold change in ions.
- Grade level
- 8–12
- Curriculum band
- Difficulty
- Beginner
- For this topic
- Time to master
- 1 week
- With daily practice
- Tools on page
- 8 on page
- Calculator + practice
Calculator
pH Input — [H⁺] mol/L
pH Summary
Breakdown
| Property | Value |
|---|---|
| log₁₀[H⁺] | -3 |
| [OH⁻] (mol/L) | 1e-11 |
| Check: 10^(−pH) | 0.001 |
Results
pH
3
Acidic — pH 3
Acidic
pH < 7
1e-11mol/L
10⁻¹⁴ / [H⁺] companion
-3
Negated to get pH
Results apply pH = −log₁₀[H⁺] for the input given. Concentration must be strictly positive.
Statistics
Live readouts from your current input — meter, bars and cards all move together.
pH
3
Acidic
Class
Acidic
pH < 7
[OH⁻]
1e-11
mol/L companion
Visualizations
pH vs neutral line
pH across concentrations
Acid–base meter
pH rides a flipped log curve — dilute right, concentrated left
Formula
A flipped base-10 log: concentrated acids read low, dilute bases read high, pure water sits at 7.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| [H⁺] | Hydrogen ions | mol/L | Concentration; must be > 0 |
| pH | pH value | pH | Negated log₁₀ of [H⁺] |
| [OH⁻] | Hydroxide ions | mol/L | 10⁻¹⁴/[H⁺] at 25 °C |
| 7 | Neutral | pH | Pure water at 25 °C |
Explanation
Sørensen flipped the log so chemists say “pH 3” instead of “0.001 molar hydrogen” — each unit is a tenfold step.
Derivation
pH = −log₁₀[H⁺] by definition. Invert: [H⁺] = 10^(−pH). Water: Kw = 10⁻¹⁴ → pH + pOH = 14.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Write the concentration in mol/L.
Take log₁₀ to count the tens.
Negate for the pH scale.
Classify against neutral 7.
Verify by exponentiating back.
Confirm the domain held.
Worked Examples
See how the calculator works with different scenarios.
Scenario
[H⁺] = 0.001 mol/L.
Inputs
Calculation
log₁₀(0.001) = −3; pH = 3
Result
pH 3 — Acidic
Interpretation
Vinegar-strength acidity.
Understanding Your Result
Your Result: pH: 3
Typical range: Stomach acid ~1.5–2 · pure water 7 · blood 7.4 · bleach ~13.
What This Means
The result is the pH of the solution with an acidic/neutral/basic verdict attached.
Why It Matters
Dosing pools, blood buffers, and soil all hinge on pH — misreading by 1 means 10× the ions.
Professional Interpretation
Pair pH with temperature (neutral shifts off 7 when hot) and confirm with [OH⁻].
Recommendations
- •Verify with 10^(−pH)
- •Remember each unit is 10×
- •Check temperature for neutrality
Learn — The Complete Guide
pH compresses ion concentrations into one flipped log. This calculator evaluates it and classifies the result.
The core concept
pH = −log₁₀[H⁺]: the minus turns tiny concentrations into friendly 0–14 numbers (for dilute solutions).
- pH flips the log so acids read low
- Each unit is a tenfold ion change
- 7 is neutral only at 25 °C
Derivations — why the formulas work
Definition — negate the log
- 1[H⁺] = h
- 2log₁₀(h)
- 3pH = −log₁₀(h)
- 4Example: −(−3) = 3 ✓
Water pair — Kw = 10⁻¹⁴
- 1[H⁺][OH⁻] = 10⁻¹⁴
- 2−log₁₀ both sides
- 3pH + pOH = 14
- 4[OH⁻] = 10⁻¹⁴/[H⁺] ✓
Where you meet this in real life
Learn this first — prerequisites in order
- 1Common logs: evaluate log₁₀ fluently
- 2Scientific notation: read 10⁻³ style
- 3Negatives: negate logs cleanly
Careers that use this daily
- •Chemist — buffers and titrations
- •Nurse — blood-gas interpretation
- •Agronomist — soil pH management
- •Brewer — mash and ferment control
Why it helps
- •Instant pH with class labeling
- •[OH⁻] companion included
- •Round-trip 10^(−pH) proof
Best practices
- •Verify with the inverse: 10^(−pH)
- •Think in tenfold steps per unit
- •Note temperature on lab reports
Limitations — what this tool does not do
- •Dilute-solution framing (activity ignored)
- •25 °C neutral = 7 assumed
- •Display rounds to 12 significant digits
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| Middle / Physical Science | 7–9 | MS-PS1 | Read pH strips · Acid/base labels · Tenfold steps |
| Chemistry | 10–12 | HS-PS1 | Compute pH/pOH · Invert to [H⁺] · Buffer basics |
Timeline — how this idea grew up
1909
Sørensen defines pH
Carlsberg brewery chemistry needs a compact acidity number.
1920s
Glass electrodes arrive
Beckman’s pH meter turns the log into a dial reading.
Today
pH everywhere
Pools, blood, soil, and food labels all speak Sørensen’s scale.
A little history
- • The “p” in pH is Danish potenz — “power” (of hydrogen).
- • Sørensen invented pH for beer; winemakers and doctors adopted it.
Exam-day checklist
- • Negate last — sign errors are #1.
- • Count the exponent for powers of 10.
- • Verify with 10^(−pH).
- • Low pH = strong acid.
Myths vs facts
Practical Tips
- •Verify: 10^(−pH) must match input
- •Low pH = more acidic
- •Each unit is a 10× change
- •Reject [H⁺] ≤ 0 immediately
- •Note temperature on reports
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 I calculate pH from [H⁺]?
What counts as acidic, neutral, or basic?
Why is each pH unit a tenfold change?
Can pH go below 0 or above 14?
How do I go backwards from pH to [H⁺]?
How do I use this pH calculator?
Related Calculators
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Related Guides
Deepen your understanding with these educational resources.
Glossary
Technical terms used in mortgage calculations explained in plain language.
- pH
- −log₁₀[H⁺]; the acidity number.
- [H⁺]
- Hydrogen-ion concentration in mol/L.
- Acidic
- pH < 7 at 25 °C.
- Neutral
- pH = 7 at 25 °C.
- Basic
- pH > 7 at 25 °C.
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 stomach is pH ~2
Gastric acid at 0.01 M H⁺ is 100,000× more acidic than pure water — digestion by brute ion force.
Blood pH barely moves
Blood holds 7.35–7.45; a 0.3 drift is a medical emergency — buffers fight every tenfold swing.
Sørensen brewed this up
pH was invented in 1909 for beer quality control at Carlsberg — chemistry’s most famous brewery export.
Rain is naturally acidic
Clean rain sits near pH 5.6 from dissolved CO₂ — “acid rain” means pushed well below that.
Compare Side by Side
Your sample between a kitchen acid and pure water — tenfold steps all the way.
| Solution | [H⁺] mol/L | pH | Class |
|---|---|---|---|
| Lemon (~pH 2) | 0.01 | 2 | Acidic |
| Your sample | 0.001 | 3 | Acidic |
| Pure water (pH 7) | 0.0000001 | 7 | Neutral |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
pH(0.001 M) = 3 (Acidic)
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)
- ▸Scientific notation (10⁻³ style)
- ▸Negative numbers (negation moves)
Mastery progress — 0%
- I can compute pH from any [H⁺]
- I can invert pH to concentration
- I can explain tenfold steps
- I can classify any reading
- 1
Week 1 · ~20 min/day
Compute pH
Goals
- Evaluate −log₁₀[H⁺] for powers of 10
- Interpolate non-powers (≈ values)
- State the domain: [H⁺] > 0
Practice
Compute pH for 10 concentrations from 1 to 10⁻¹⁴.
- 2
Week 2 · ~25 min/day
Classify + invert
Goals
- Label acidic/neutral/basic instantly
- Invert pH → [H⁺] with 10^(−pH)
- Convert pH ↔ pOH with 14
Practice
Round-trip 5 readings both directions.
- 3
Week 3 · ~20 min/day
pH in the wild
Goals
- Read pool strips quantitatively
- Explain buffers in tenfold terms
- Dilution-shift pH by logs
Practice
Test one household liquid and justify its label.
Exam tips
- •Negate last — sign errors are #1.
- •Count the exponent for powers of 10.
- •Verify with 10^(−pH).
Practice Problems
0/5 solvedTry each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.
[H⁺] = 0.001 M. Find pH.
[H⁺] = 10⁻⁷ M. Find pH.
pH 13 → [H⁺]?
How much stronger is pH 2 than pH 5?
Dilute pH-3 acid 100×. New pH?
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 |
|---|---|---|
| pH = −log₁₀[H⁺] | Master formula | 0.001→pH 3 |
| [H⁺] = 10^(−pH) | Invert | pH 3→0.001 M |
| pH + pOH = 14 | Water pair (25 °C) | pH 3→pOH 11 |
| ΔpH 1 = 10× ions | Tenfold steps | pH2 vs 5→1000× |
| pH 7 = neutral | Water anchor | <7 acid, >7 base |
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Page Information
2026-09-29
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