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A logarithmic scale places a positive quantity $x$ according to a logarithm, so equal ratios occupy equal distances. For example, $1,2,4,8$ are equally spaced under the natural logarithm because every successive difference is $\ln 2$. An ordinary linear scale instead gives equal distances to equal differences: $1,2,3,4$.

For a base $b>1$ and a positive reference $x_0$, define the axis positionThe reference has the same units as $x$, making $x/x_0$ a dimensionless quantity. The distance between two positions isMultiplying both values by the same positive factor preserves that distance. Replacing $x_0$ by $c{x}_0$ translates every position by $-{\log }_{b}c$; changing the base rescales positions through ${\log }_{b}z=\ln z/\ln b$. Neither change alters which ratios receive equal gaps.

In base ten, the values $1,10,100,1000$ occupy positions $0,1,2,3$ when $x_0=1$. The differences between neighboring values grow from nine to ninety to nine hundred, but every step spans one order of magnitude. The values $2,20,200$ occupy positions $0.30103,1.30103,2.30103$: the same spacing shifted by ${\log }_{10}2$. This lets one plot small and large positive quantities together while keeping equal proportional changes equally visible.

A logarithmic scale is a rule for position, not an instruction to relabel the tick marks with their logarithms. A mark labelled $100$ still means the original value one hundred. Ordinary real logarithms exclude zero and negative inputs. Omitting such values or clipping them to a plotting boundary does not give them genuine logarithmic positions; the plotting choice needs to be clear.^[ref]

For positive quantities, a difference of one order of magnitude means a factor of ten; a difference of $n$ orders of magnitude means a factor of $10^n$. The logarithmic difference ${\log }_{10}(x_2/x_1)$ records this comparison, without subtracting the quantities themselves. For example, $1000$ is three orders of magnitude above one, regardless of a common change of units. Roughly assigning a single value to an order of magnitude also requires a rounding convention; powers of ten are the reference levels.