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Study Guides > College Algebra CoRequisite Course

Introduction to Properties of Logarithms

Learning Outcomes

By the end of this lesson, you will be able to:
  • Rewrite a logarithmic expression using the power rule, product rule, or quotient rule.
  • Expand logarithmic expressions using a combination of logarithm rules.
  • Condense logarithmic expressions using logarithm rules.
  • Expand a logarithm using a combination of logarithm rules.
  • Condense a logarithmic expression into one logarithm.
  • Rewrite logarithms with a different base using the change of base formula.
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 [latex]a[/latex] is the concentration of hydrogen ion in the solution.

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

[latex]-\mathrm{log}\left(\left[{H}^{+}\right]\right)[/latex] is equal to [latex]\mathrm{log}\left(\frac{1}{\left[{H}^{+}\right]}\right)[/latex] due to one of the logarithm properties we will examine in this section.

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  • Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..
  • College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].