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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 aa is the concentration of hydrogen ion in the solution.

pH=−log([H+])=log(1[H+])\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}

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

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