Logarithms: History and Common Bases
Logarithms were originally invented by John Napier (1515-1617) to aid in arithmetical computations at a time when modern day calculators were not in use. In the present day, logarithms have many uses in disciplines as different as economics, computer science, engineering and natural sciences. Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting.
While any positive number can be used as the base of a logarithm, not all logarithms are equally useful in practice. Some bases have more applications than others. Out of the infinite number of possible bases, three stand out as particularly useful. These are
The Common Logarithm
A logarithm with a base of
The Natural Logarithm
A logarithm with a base of
The Binary Logarithm
A logarithm with a base of
Uses of Logarithms
The list below highlights only some of the many uses of logarithms in the present day.
1. The magnitude of an earthquake (M) can be determined based on the logarithm of an intensity measurement from a seismograph (I):
2. The common logarithm is used in calculating the safety index which helps determine how safe certain activities are by determining how likely people are to die from them. For example,
3. pH is an abbreviation for power of hydrogen. The pH scale measures how acidic or basic a substance is. It ranges from
Pure water contains a hydrogen ion concentration of
4. The entropy
5. Logarithms occur in definitions of the dimension of fractals. Fractals are geometric objects that are self-similar: small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle can be covered by three copies of itself, each having sides half the original length. This makes the Hausdorff dimension of this structure
Sierpinski triangle
The Sierpinski triangle can be covered by three copies of itself, over, and over, and over.
6. Logarithms are related to musical tones and intervals. In equal temperament, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch, of the individual tones. For example, the note A has a frequency of
7. Natural logarithms are closely linked to counting prime numbers (