How to calculate pH with logarithms: formula and examples

To calculate pH from a known hydrogen-ion concentration, take its base-10 logarithm and change the sign. For example, [H⁺] = 10⁻³ mol/L gives pH 3 under the ideal concentration approximation. When the concentration includes a coefficient, such as 2 × 10⁻⁵, the logarithm of that coefficient matters too.

In an IMAT chemistry question, a wrong pH answer can come from a logarithm or minus-sign error. The steps below help you separate that maths error from an acid–base chemistry error.

What pH actually says

pH = −log₁₀[H⁺]

This is the concentration approximation used below, with concentrations in mol/L and activity effects neglected. The IUPAC definition uses hydrogen-ion activity. Concentrated solutions can deviate from this simple model, and pH is not restricted to the range 0–14.

Read backwards, it says something useful:

[H⁺] = 10^−pH

So pH 3 means [H⁺] = 10⁻³ mol/L. Not “somewhat acidic” — approximately one thousandth of a mole per litre under this model.

01234567891011121314

← more acidic neutral more basic →

pH 7 [H⁺] = 10⁻⁷ mol/L

the neutral point

• Pure water at 25 °C

Neutral does not mean "no ions". Water self-ionises: [H⁺] = [OH⁻] = 10⁻⁷ mol/L. Neutral means the two are equal, not absent.

Four useful log rules

Four, and you have met all of them in maths:

Rule	Example
log(10ⁿ) = n	log(10⁻⁵) = −5
log(a × b) = log a + log b	log(2 × 10⁻⁵) = log 2 + log 10⁻⁵
log(a ÷ b) = log a − log b	log(5/2) = log 5 − log 2
log 1 = 0	pH of [H⁺] = 1 is 0

Two useful approximate values: log 2 ≈ 0.3 and log 3 ≈ 0.48. With those you can handle most exam numbers without a calculator.

How to calculate pH without a calculator

For [H⁺] = a × 10⁻ⁿ mol/L, the concentration approximation gives pH = n − log₁₀ a. First identify the exponent, then subtract the logarithm of the coefficient. With 1 ≤ a < 10, the pH lies above n − 1 and at or below n.

Worked example: [H⁺] = 2 × 10⁻⁵ mol/L

What is the pH of a solution with [H⁺] = 2 × 10⁻⁵ mol/L?

Apply the definition, then split the product:

pH = −log(2 × 10⁻⁵) = −(log 2 + log 10⁻⁵)
= −(0.3 − 5) = 4.7

Notice the direction. The answer is 4.7, not 5.3 — a concentration higher than 10⁻⁵ means a lower pH. If your answer moved the wrong way, you dropped a minus sign. Checking the direction helps catch that error.

Why “ten times” matters more than “one unit”

Because the scale is logarithmic, differences multiply rather than add:

• pH 3 versus pH 4 → 10× the hydrogen-ion concentration
• pH 3 versus pH 6 → 1 000× the hydrogen-ion concentration
• pH 2 versus pH 7 → 100 000× the hydrogen-ion concentration

A question that says “how many times more acidic” is testing exactly this. The answer is never the difference between the numbers; it is ten to the power of that difference.

For a fully dissociated monoprotic strong acid, tenfold dilution raises pH by approximately one while the acid supplies much more H⁺ than water does. Near neutrality, water autoionisation matters and that shortcut fails. Dilution with pure water approaches neutral conditions, approximately pH 7 at 25 °C, rather than turning the acid into a base.

The other side of the scale

The same trick applies to bases:

pOH = −log[OH⁻] and pH + pOH = 14

That relationship comes from the ion product of water, K_w = [H⁺][OH⁻] = 10⁻¹⁴ at 25 °C. Take logs of both sides and the 14 appears. It holds at 25 °C — a detail worth remembering, since K_w changes with temperature.

So a solution with [OH⁻] = 10⁻³ has pOH 3, and therefore pH 11.

The five confusions that cost marks

1. “pH 6 is twice as acidic as pH 3.” It is a thousand times less acidic.
2. Losing the minus sign, and landing on 5.3 where 4.7 was correct.
3. “[H⁺] = 2 × 10⁻⁵ means pH 5.” It means pH 4.7 — the coefficient shifts it.
4. “Enough dilution turns acid into base.” Dilution with pure water approaches neutrality: approximately pH 7 at 25 °C.
5. Reaching for pOH and forgetting the 14, or applying it at temperatures where K_w is no longer 10⁻¹⁴.

When the chemistry comes first

For a weak acid or a buffer, the supplied acid concentration is not generally the hydrogen-ion concentration. Identify the equilibrium or neutralisation step before taking the logarithm. See acids, bases and buffers.

The wider point

If several chemistry questions across different topics keep going wrong — pH, titration curves, buffer capacity — check whether they share a maths dependency before concluding that chemistry is the problem. Fixing one prerequisite can unlock three topics at once, which is a far better use of an afternoon than re-reading a chapter you already understood.