Logarithmic scales
Read the quantity labels and units before interpreting a distance. Equal distances represent equal ratios on the same logarithmic scale
- Decades One base-10 decade multiplies a positive quantity by ten
- Direction Moving left divides by the same factor that moving right multiplies by
- Small values Negative exponents represent small positive quantities, not negative quantities
- Zero An ordinary logarithmic scale has no finite position for zero or a negative quantity
- Midpoints Use the same factor from the lower endpoint to the midpoint and from the midpoint to the upper endpoint. This gives the geometric midpoint
- Applied scales Read coefficients, signs, units and reference values. Named scales need not share the same numerical conversion
Worked example
Two samples have masses 0.00001 g and 0.01 g. Find the factor from the smaller to the larger and the number of base-10 decades
- \(\displaystyle \begin{aligned}0.00001&=10^{-5}\\0.01&=10^{-2}\end{aligned}\)Express both quantities using the same unit
- \(\displaystyle (-2)-(-5)=3\)The exponents differ by three, so the scale distance is three decades
- \(\displaystyle \frac{0.01}{0.00001}=10^3=1000\)The larger mass is 1000 times the smaller
Common mistakes
Difficulty levels
Questions about this topic
Must I know logarithm laws first?+
No. This lesson starts with powers of ten and equal multiplicative steps. Logarithm notation names the exponent; broader laws are separate extension content
Can the scale include negative quantities?+
An ordinary real logarithmic scale requires positive quantities. A negative exponent is allowed because its power of ten is still positive
Does a logarithmic display change the data?+
No. It changes positions and distances. Always retain the original quantity, units, uncertainty and context
Are decibels a direct measure of perceived loudness?+
The stated exercise compares physical intensity. Perception and hearing safety depend on other factors and are not conclusions of this calculation
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 3: Algebra and indices