Year 10AC9M10M02

Logarithmic Scales

Interpret and use logarithmic scales in applied contexts involving small and large quantities and change.

Some quantities range over so many powers of ten that an ordinary axis is useless. A logarithmic scale fixes that by making equal distances mean equal multiplications rather than equal additions.

Method

Each step multiplies

1,\ 10,\ 100,\ 1\,000,\ 10\,000

On an ordinary scale each step adds. On a log scale each step multiplies by ten.

Equal distance means equal ratio

The gap from 1 to 10 is the same size as the gap from 1000 to 10 000. Both are one multiplication by ten.

A difference of $n$ means $\times 10^{n}$

\text{5 to 7} \ \Rightarrow\ 10^{2} = 100\times

Two steps up a log scale is a hundred times bigger, not twice as big.

Zero and negatives have no place

You cannot reach zero by multiplying, so a log scale never shows it.

Worked example

e.g.

\text{An earthquake of magnitude 7 compared with one of magnitude 5}

Find the difference in the scale readings. $7 - 5 = 2$
Each unit is one multiplication by ten, so raise ten to that difference. $10^{2}$
State the comparison as a multiplier, never as a subtraction. $100 \text{ times larger}$
Check you have not read it as ordinary numbers. $\text{not } 7 \div 5 = 1{,}4 \text{ times}$
For a value between marks, work with the ratio. $\text{magnitude } 6{,}5 \text{ is } 10^{1{,}5} \approx 32 \text{ times a magnitude 5}$
Answer in the language of the context. $\text{The 7 releases about a hundred times the energy of the 5.}$

Practice

A difference on the scale is always a power of ten.

\text{How many times louder is } 80 \text{ dB than } 60 \text{ dB, in scale terms?}

Answer

10^{2} = 100 \text{ times}

\text{A pH of 4 compared with a pH of 6}

Answer

10^{2} = 100 \text{ times more acidic}

\text{Why can a log scale not show zero?}

Answer

\text{No power of ten equals zero, so zero has no position.}

\text{Two points are the same distance apart on a log axis: 2 to 20, and 500 to 5000. Same ratio?}

Answer

\text{Yes — both are } \times 10

\text{A magnitude 8 compared with a magnitude 5}

Answer

10^{3} = 1000 \text{ times}