pH & Logarithms — The Scale of Acidity
The pH Scale
Every liquid you encounter — coffee, blood, pool water, battery acid — has a pH determined by a logarithm.
$$\text{pH} = -\log_{10}[\text{H}^+]$$
where $[\text{H}^+]$ is the hydrogen ion concentration in moles per liter (M).
The negative logarithm compresses an enormous range of concentrations into a tidy 0–14 scale:
| Substance | [H⁺] (M) | pH |
|---|---|---|
| Battery acid | $1.0$ | 0 |
| Lemon juice | $10^{-2}$ | 2 |
| Coffee | $10^{-5}$ | 5 |
| Pure water | $10^{-7}$ | 7 |
| Blood | $3.98 \times 10^{-8}$ | 7.4 |
| Bleach | $10^{-13}$ | 13 |
Why Logarithms?
The concentration of H⁺ spans 14 orders of magnitude — from $1$ to $10^{-14}$. Without logarithms, comparing battery acid to bleach would mean comparing $1$ to $0.00000000000001$. The log scale makes this manageable.
Each pH unit represents a 10× change in acidity. A substance with pH 3 is 10 times more acidic than pH 4, and 100 times more acidic than pH 5.
If $[\text{H}^+] = 3.16 \times 10^{-4}$ M, what is the pH?
$$\text{pH} = -\log_{10}(3.16 \times 10^{-4}) = -(\log 3.16 + \log 10^{-4})$$
$$= -(0.5 + (-4)) = -(0.5 - 4) = 3.5$$
The solution has pH 3.5 — quite acidic, like orange juice.
The logarithm is one of the most practical functions in science. It turns multiplicative relationships into additive ones and makes enormous ranges human-readable. Every time you see a "scale" (pH, decibels, Richter), a logarithm is behind it.