Disinfection is written in logs because percentages lie about the thing that matters. $\mathrm{LR} = \log_{10}\!\left(\dfrac{N_0}{N}\right)$ — read aloud L R equals log base ten of N-nought over N. $N_0$ — N-nought — is the count before the process, organisms per 100 mL; $N$ is the count after, same units; and $\mathrm{LR}$ is the log reduction, a bare number of base-10 logs. Before goes on top. Put it the other way and the answer arrives negative, which is the equation telling you so.

The translation to percent is $P = 1 - 10^{-\mathrm{LR}}$, where $P$ is the fraction inactivated, quoted as a percentage. Learn the ladder and you will rarely need the algebra: 1-log is 90%, 2-log is 99%, 3-log is 99.9%, 4-log is 99.99%. One nine per log, forever.

Now the error this lesson exists to kill. 99.9% is 3-log. It is never 99.9-log. That claim appears in real reports, and it describes a kill of one part in $10^{99.9}$ — more organisms than there are atoms in the observable universe. When you see a percentage and a log figure in the same sentence, check that they are the same statement said twice, not two credits added together.

Why logs at all? Because logs add across barriers and percents do not. Two barriers of 2-log each give 4-log overall — 99.99%, not 198% and not 99%. And because 99% and 99.99% look nearly identical on a page while differing by a factor of a hundred in survivors. The reading worth keeping in your head is the survivor: 3-log means one in a thousand lived.

Behind the measurement sits the model. Chick–Watson, $\mathrm{LR} = k\,C^{\,n}\,t$, predicts the reduction instead of counting it: $k$ is a fitted rate coefficient for that organism and disinfectant, $C$ the residual in mg/L, $t$ the contact time in minutes, and $n$ — the dilution coefficient — is Watson's exponent, near 1 for free chlorine. Set $n = 1$ and it becomes $\mathrm{LR} = k \times \text{CT}$: the regulatory shortcut is this equation with its exponent pinned. It is an idealisation — real survival curves show a shoulder at the start and a stubborn tail at the end — but it is the reason CT works at all.

Last caution, and it is a discipline rather than a formula. A non-detect is not a zero. Report it as the detection limit and you get the MINIMUM log reduction your data supports; treat it as zero and you have claimed infinite reduction, which is how most inflated numbers get made.