Logarithmic scales are used when numbers cover a huge range, from tiny fractions to enormous values. Instead of spacing values by equal additions, a logarithmic scale spaces them by equal multiplication factors. This makes it possible to compare earthquakes, sound intensity, acidity, populations, and scientific measurements on one readable graph.
The main idea is that each step represents an order of magnitude, often a factor of 10.
Key Facts
- On a base 10 logarithmic scale, equal spacing means multiplying by 10 each step.
- log10(10) = 1, log10(100) = 2, and log10(1000) = 3.
- If log10(x) increases by 1, x becomes 10 times larger.
- A value of 10^n has order of magnitude n on a base 10 scale.
- Decibels use a logarithmic scale: dB = 10 log10(I / I0).
- pH is logarithmic: pH = -log10([H+]), so a pH decrease of 1 means 10 times more hydrogen ion concentration.
Vocabulary
- Logarithm
- A logarithm tells which exponent is needed to produce a given number from a chosen base.
- Logarithmic scale
- A scale where equal distances represent equal multiplication factors rather than equal additions.
- Base
- The base is the number being raised to a power in a logarithm, such as 10 in log10(x).
- Order of magnitude
- An order of magnitude is a factor of 10 difference between values.
- Decibel
- A decibel is a logarithmic unit used to compare sound intensity or other power ratios.
Common Mistakes to Avoid
- Treating equal spaces as equal additions is wrong because on a logarithmic scale equal spaces represent equal ratios, such as multiplying by 10 each step.
- Reading halfway between 10 and 100 as 55 is wrong because the midpoint on a base 10 log scale is about 31.6, since it represents the geometric middle.
- Forgetting the base is wrong because log2(x), log10(x), and ln(x) use different reference numbers and give different numerical results.
- Saying a pH change from 4 to 3 is a small linear change is wrong because pH is logarithmic, so the solution has 10 times greater hydrogen ion concentration.
Practice Questions
- 1 On a base 10 logarithmic scale, points are marked at 1, 10, 100, 1000, and 10000. How many equal log-scale steps are there from 10 to 10000, and by what total factor does the value increase?
- 2 A sound has intensity I = 1000000 I0. Using dB = 10 log10(I / I0), find the sound level in decibels.
- 3 Two earthquakes have Richter magnitudes 5 and 7. Explain why the magnitude 7 earthquake represents much larger measured wave amplitude than the magnitude 5 earthquake.