# Chapter 9.6: 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.

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 is the concentration of hydrogen ion in the solution

The equivalence of and is one of the logarithm properties we will examine in this section.

### Using the Product Rule for Logarithms

Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy to prove.

For example, since And since

Next, we have the inverse property.

For example, to evaluate we can rewrite the logarithm as and then apply the inverse property to get

To evaluate we can rewrite the logarithm as and then apply the inverse property to get

Finally, we have the one-to-one property.

We can use the one-to-one property to solve the equation for Since the bases are the same, we can apply the one-to-one property by setting the arguments equal and solving for

But what about the equation The one-to-one property does not help us in this instance. Before we can solve an equation like this, we need a method for combining terms on the left side of the equation.

Recall that we use the *product rule of exponents* to combine the product of exponents by adding: We have a similar property for logarithms, called the **product rule for logarithms**, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.

Given any real number and positive real numbers and where we will show

Let and In exponential form, these equations are and It follows that

Note that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of any number of factors. For example, consider Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:

### The Product Rule for Logarithms

The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.

### How To

**Given the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.**

- Factor the argument completely, expressing each whole number factor as a product of primes.
- Write the equivalent expression by summing the logarithms of each factor.

### Using the Product Rule for Logarithms

Expand

## Show Solution

We begin by factoring the argument completely, expressing as a product of primes.

Next we write the equivalent equation by summing the logarithms of each factor.

### Try It

Expand