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Study Guides > College Algebra

Introduction to Logarithmic Properties

LEARNING OBJECTIVES

By the end of this lesson, you will be able to:
  • 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.
Testing of the pH of hydrochloric acid. The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan)
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 a is the concentration of hydrogen ion in the solution

pH=−log([H+]) =log(1[H+])\begin{array}{l}\text{pH}=-\mathrm{log}\left(\left[{H}^{+}\right]\right)\hfill \\ \text{ }=\mathrm{log}\left(\frac{1}{\left[{H}^{+}\right]}\right)\hfill \end{array}

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

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