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Properties of Logarithms

Logarithmic Properties

Learning Objectives

In this section, you will:

  • Use the product rule for logarithms.
  • Use the quotient rule for logarithms.
  • Use the power rule for logarithms.
  • Expand logarithmic expressions.
  • Condense logarithmic expressions.
  • Use the change-of-base formula for logarithms.


Figure 1 The pH of hydrochloric acid is tested with litmus paper.

In chemistry, pH is used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14. Substances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be alkaline. Our bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is acidic and what is alkaline, consider the following pH levels of some common substances:

  • Battery acid: 0.8
  • Stomach acid: 2.7
  • Orange juice: 3.3
  • Pure water: 7 (at 25° C)
  • Human blood: 7.35
  • Fresh coconut: 7.8
  • Sodium hydroxide (lye): 14

To determine whether a solution is acidic or alkaline, we find its pH, which is a measure of the number of active positive hydrogen ions in the solution. The pH is defined by the following formula, where \(H^+\) is the concentration of hydrogen ion in the solution

\(pH=−log([H^+])\)

\(=log \left(\frac{1}{[H^+]} \right)\)

The equivalence of \(−log([H^+])\) and \(log \left(\frac{1}{[H^+]} \right)\) is one of the logarithm properties we will examine in this section.

Source: Rice University, https://openstax.org/books/college-algebra/pages/6-5-logarithmic-properties
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