Maths: Why a Small pH Shift Means a Big Change

Logarithmic scales let us compress vast ranges of physical quantities into manageable numbers. The pH scale is a classic example: pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in mol L⁻¹. Because hydrogen ion concentrations can span many orders of magnitude, a logarithmic scale turns multiplicative changes into additive ones, making it far easier to compare solutions. This matters because a small shift in pH corresponds to a dramatic change in concentration. Applying log laws to pH = −log₁₀(4.5 × 10⁻⁵) splits the expression into log₁₀ 4.5 plus log₁₀ 10⁻⁵, showing how the exponent and coefficient separate. When two solutions differ by a fixed number of pH units, subtracting their pH expressions gives log₁₀ of the concentration ratio, so that difference becomes the exponent of 10. This relationship between additive pH changes and multiplicative concentration changes is the heart of the concept.