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II-02: Base-10 logarithms and inverse logarithms

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Module II-02 · Recommended before: Before pH and pKa

Base-10 logarithms and inverse logarithms

A logarithm answers which power produces a number. For pH, every factor-of-ten change in hydronium concentration changes pH by one unit.

Learn the method

pH = −log₁₀[H₃O⁺]

  1. Recognize log₁₀(10ⁿ)=n.
  2. Apply pH = −log₁₀[H₃O⁺].
  3. Reverse with [H₃O⁺] = 10⁻ᵖᴴ.
  4. Check signs and significant decimal places in pH.

Common mistake: Using natural log instead of base-10 log changes pH.

Worked example

Problem: Compute pH when [H₃O⁺] = 1.0×10⁻³ M.

Step-by-step solution: pH = −log₁₀(1.0×10⁻³) = 3.00.

Check the units and whether the answer is reasonable before continuing.

More applications of this mathematics

Try each extension before expanding its solution.

Reverse pH calculation

pH 6.20 corresponds to [H₃O⁺]=10⁻⁶·²⁰=6.3×10⁻⁷ M.

Logarithm laws

log₁₀(AB)=log₁₀A+log₁₀B. In contrast, log₁₀(A+B) is not log₁₀A+log₁₀B.

Independent practice — five questions

Suggested mastery threshold: four out of five. Incorrect answers display a worked explanation after submission.

Choose one response for each question, then select “Check my answers.” No timer. Hints are optional. You can retry freely. Questions are formative, not proctored.

Question 1. Find pH when [H₃O⁺] = 1.0 × 10^-6 M.
Hint for question 1

The negative sign makes pH positive for concentrations below 1 M.

Question 2. Find pH when [H₃O⁺] = 1.0 × 10^-7 M.
Hint for question 2

The negative sign makes pH positive for concentrations below 1 M.

Question 3. Find pH when [H₃O⁺] = 1.0 × 10^-8 M.
Hint for question 3

The negative sign makes pH positive for concentrations below 1 M.

Question 4. Find pH when [H₃O⁺] = 1.0 × 10^-9 M.
Hint for question 4

The negative sign makes pH positive for concentrations below 1 M.

Question 5. Find pH when [H₃O⁺] = 1.0 × 10^-10 M.
Hint for question 5

The negative sign makes pH positive for concentrations below 1 M.

Supporting reference

OpenStax Chemistry 2e / NIST — reference for this subject. This lesson’s wording, examples and questions are original.

W3C WCAG 2.2 accessibility guidance.

Applied practice: pH and acid-base logarithms

These ungraded questions extend the scored five-question check. Work each problem before opening its solution; progress records are not changed.

  1. Challenge 1. A solution has pH = 4.25. Calculate [H₃O⁺].

    Show worked solution for challenge 1

    [H₃O⁺] = 10⁻ᵖᴴ = 10⁻⁴·²⁵ = 5.6 × 10⁻⁵ M. The two decimal places in pH correspond to two significant figures in concentration.

  2. Challenge 2. A weak acid has pKₐ = 4.76. Calculate Kₐ.

    Show worked solution for challenge 2

    Kₐ = 10⁻ᵖᴷᵃ = 10⁻⁴·⁷⁶ ≈ 1.7 × 10⁻⁵.

IntoChemistry original instructional content, 2026. Formative feedback is not an official course grade. Sources are linked for independent reading. Accessibility testing with your institution and assistive technology is still recommended.