The logarithmic scale

When the values of a quantity span many orders of magnitude, a linear scale is useless: one adopts a logarithmic scale, where each tick corresponds to a power of the base.

Definition — Logarithmic scale

The logarithmic scale (in base $10$) of a positive quantity $x$ is the number $L = \log_{10} x$. On the logarithmic axis the ticks correspond to the powers of $10$: $1, 10, 100, 1000, \ldots$ are equally spaced (respective distances $0,1,2,3,\ldots$).

Comparison between a linear scale and a logarithmic scale: on the logarithmic axis the powers of $10$ are equally spaced.

Property — Transforming products into sums

On the logarithmic scale, the multiplications in the linear world become additions: $\log(xy) = \log x + \log y$. For this reason a semi-logarithmic graph turns exponential curves into straight lines: if $y = A\cdot 10^{kx}$, then $\log y = \log A + kx$, that is $\log y$ is a linear function of $x$.

Example — Semi-logarithmic paper

Bacterial growth $N(t) = N_0\cdot 2^{t/T}$ (doubling every $T$ minutes) appears as a straight line on semi-logarithmic paper ($\log N$ vs $t$): $\log N = \log N_0 + (t/T)\log 2$. The slope $\log 2 / T \approx 0{,}301/T$ lets one read $T$ off the graph.

Links

Topics: Logarithmic function
Concepts: Logarithm · Logarithmic scale
Functions: Logarithmic function
Methods: Logarithmic scale
Skills: Interpret graphs · Model