Definition

1. Adverb:
   • In a manner relating to or using logarithms: Describes an action, process, or relationship that involves, follows, or is measured on a logarithmic scale.
   • At a rate proportional to the logarithm of a value: Often used to describe growth, decrease, or the spacing of values that follow multiplicative rather than additive patterns.

Usage and Examples

• Adverb:
  • The population of the bacteria increased logarithmically during the initial phase. (It grew at a rate proportional to the logarithm of its size.)
  • The data is plotted logarithmically on the vertical axis to better visualize the wide range of values. (The scale on the axis is based on logarithms.)
  • Sound intensity is measured logarithmically in decibels. (The measurement uses a logarithmic function to relate perceived loudness to physical intensity.)

Advanced Usage

• "to scale logarithmically": to use a logarithmic scale for measurement or representation.
  • The graph's y-axis scales logarithmically, so each increment represents a tenfold increase.
• "to increase/decrease logarithmically": to grow or diminish at a rate characterized by a logarithmic function.
  • The algorithm's efficiency improves logarithmically with the size of the dataset, which is very favorable.

Variants and Related Words

• Logarithmic (adjective): Of, relating to, or expressed in terms of logarithms.
  • A logarithmic scale is essential for charting earthquake magnitudes.
• Logarithm (noun): A quantity representing the power to which a fixed number (the base) must be raised to produce a given number.
  • The logarithm of 100 to base 10 is 2.

Synonyms

• Exponentially (Note: This is a related but distinct concept. "Exponentially" describes growth by a constant factor, while "logarithmically" often describes the inverse relationship or a rate of change relative to a logarithm.)
• On a log scale: Using a logarithmic scale for measurement or display.

Related Phrases and Technical Usage

• "plotted logarithmically": graphed using logarithmic scales on one or more axes.
  • When the data is plotted logarithmically, the exponential trend appears as a straight line.
• "logarithmically spaced": arranged so that the ratio between consecutive values is constant, not the difference.
  • The frequency bands in the audio analyzer are logarithmically spaced to match human hearing.

Adverb

1. in a logarithmic manner
   • data plotted logarithmically with respect to time is shown in Figure 2