Logarithms — quick reminder

Before linearising a law, logarithms are needed. The base-10 logarithm of $x$, written $\log_{10} x$ (or simply $\log x$), is the exponent to which 10 must be raised to obtain $x$:

$\log_{10} x = y \iff 10^{y} = x$

A few values worth remembering: $\log_{10} 100 = 2$, because $10^2 = 100$; $\log_{10} 0{,}001 = -3$, because $10^{-3} = 0{,}001$; and $\log_{10} 1 = 0$, because $10^{0} = 1$. The logarithm grows extremely slowly: multiplying $x$ by $10$ always adds just $1$ to the logarithm. It is this compression of scales that lets a logarithmic graph fit, on the same sheet, numbers ranging from $1$ to a million.

The three fundamental properties

$\log(AB) = \log A + \log B \qquad \log\!\left(\frac{A}{B}\right) = \log A - \log B \qquad \log A^{n} = n\,\log A$

The intuition is all here: logarithms turn products into sums and powers into multiplications. A relation involving products and powers — that is, almost every non-linear physical law — becomes, once logarithms are taken, a relation made up purely of sums and proportionalities: in other words, a straight line. This is exactly what makes logarithms the natural tool for recognising and measuring exponential laws and power laws.

In practice

If you see a product or an exponent in a formula, try taking its logarithm: the formula almost always “straightens out” into something linear, much easier to read on a graph.

Links

Topics: Experimental method Skills: Use of logarithmic scales

Related exercises: Worked exercise — Bouncing ball (semilog) · Problem — Bacteria and semilog graph · Problem — Fermi estimate, measurements of g