Properties of Logarithms

Before diving into solving logarithmic and exponential equations, it is helpful to know the properties of logarithms because they can help you out of tricky situations. In this section, you will learn the algebraic properties of logarithms, including the power, product, and quotient rules.

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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