M1.04: Why We Use Logarithms

Section 2: Compressing the range of values by using logarithms

Many types of measurements cover an extremely wide range of values. An example is the energy released in earthquakes — there is a 100-billion-times difference between the energy of the smallest earthquake that a seismograph can measure and the largest ones that occur. In chemistry, concentrations of hydroxyl ion can vary by more than 100 trillion times between strong alkalis and strong acids.

Thus the same dataset might contain the relatively big value 4,200,000,000, the intermediate one 3,100, and the relatively small one 0.00025. In a regular graph of such a dataset, there is no scale that will show the data well—either the biggest number will be far off the scale at the top, or the middle number (which is less than a millionth of the big number) will be at the bottom and indistinguishable from the smallest number even though it is more than ten million times bigger.

The numbers above can be expressed as 4.2×109, 3.1×103, and 2.5×10-4, a style called scientific notation because it is used by scientists who often need to deal with very large or very small numbers. The idea of base-10 logarithms (also called “common” logarithms) carries this idea further by using decimal fractions in the exponents so that the initial number is not needed. Since 4,200,000,000109.623, we say that 9.623 is the logarithm of 4,200,000,000. Similarly, the logarithm of 3,100 is about 3.491 and that of 0.00025 is about –3.602 (all numbers between 1 and 0 have negative logarithms). So when using logarithms, the original range from 4,200,000,000 to 0.00025 becomes a compressed range from 9.623 to –3.602. On this scale, the inter­mediate value of 3.491 (the logarithm of 3,100) can be easily distinguished from both of the other values.

This is how our sense of hearing works. A whisper is a billion times less intense (in total energy into our ears) than a rock concert. In order to handle this range of input, our senses have evolved so that our perceived response to a stimulus is approximately proportional to the logarithm of its intensity. The use of logarithmic scales in mathematics and technology is a way of using this same tactic to deal with any range of numerical values where the ratio of the largest to the smallest (the dynamic range) is large.

The base of a logarithm does not have to be 10 (although only base-10, or “common”, logarithms are used in this course and in most application areas). The non-10 bases for traditional logarithmic scales are typically those that make the results into convenient numbers (e.g., between 2 and 10 for earthquakes, between 1 and 6 for star brightness). A base of 2 is used in music because notes with a frequency ratio of 2 are harmonious.

Examples of logarithmic measurement scales

  • Richter scale for earthquakes—each increase of one level means 32 times more energy
  • pH for acidity/alkalinity—each increase of 1 in pH means 10 times more hydroxyl ions.
  • Brightness of stars—each increase of 1 in stellar magnitude means a star is 2.51 times dimmer
  • Sound—on the decibel scale, each bel (= 10 decibels) means the sound is 10 times more intense
  • Music octaves—each one-octave increase in musical pitch means that vibration frequency doubles

[Optional additional information about logarithm bases: Any positive number except 1 can be used as a logarithmic base, resulting in logarithms that differ from “common” base-10 logarithms by a ratio equal to the common logarithm of the base chosen. Base-2 “binary” logarithms (3.322 times larger than common logarithms) are used mainly in computer-related fields. “Natural” logarithms (2.303 times larger than common logarithms) are used in calculus and based on the special number called e (approximately 2.71828). Natural logarithms are symbolized as “LN” on calculators, and calculated with the LN function in spreadsheets. On calculators, the LOG key computes base-10 logarithms, but spreadsheets use the LOG10 function for that purpose and use the LOG function only when the user is specifying which base to use. Thus the spreadsheet formula “=LOG(16,2)” evaluates to 4. This specify-the-base spreadsheet LOG function is useful for answering questions such as “How many years would it take for an investment at a 5% growth rate to double?” Since mathematically this is the same question as “What value of x will make (1.05)x equal to 2?”, the answer 14.2 years is computed by the formula “=LOG(2,1.05)”.]