## 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.

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

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

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