section_path: Applications/Logarithmic scale
section_name: Logarithmic scale
paragraph_text: Logarithmic scale
  
  Scientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale. For example, the decibel is a unit of measurement associated with logarithmic-scale quantities. It is based on the common logarithm of ratios —10 times the common logarithm of a power ratio or 20 times the common logarithm of a voltage ratio. It is used to quantify the attenuation or amplification of electrical signals, to describe power levels of sounds in acoustics, and the absorbance of light in the fields of spectrometry and optics. The signal-to-noise ratio describing the amount of unwanted noise in relation to a (meaningful) signal is also measured in decibels. In a similar vein, the peak signal-to-noise ratio is commonly used to assess the quality of sound and image compression methods using the logarithm.
  
  The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the moment magnitude scale or the Richter magnitude scale. For example, a 5.0 earthquake releases 32 times (10) and a 6.0 releases 1000 times (10) the energy of a 4.0. Apparent magnitude measures the brightness of stars logarithmically. In chemistry the negative of the decimal logarithm, the decimal cologarithm, is indicated by the letter p. For instance, pH is the decimal cologarithm of the activity of hydronium ions (the form hydrogen ions H take in water). The activity of hydronium ions in neutral water is 10 mol·L, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 10 of the activity, that is, vinegar's hydronium ion activity is about 10 mol·L.
  
  Semilog (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, exponential functions of the form f (x) = a · b appear as straight lines with slope equal to the logarithm of b. Log-log graphs scale both axes logarithmically, which causes functions of the form f (x) = a · x to be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzing power laws.