Chapter Five - Louisiana State University



Section 5.4 Properties of Logarithms

Objectives

1. Using the Product Rule, Quotient Rule and Power Rule for Logarithms

2. Expanding and Condensing Logarithmic Expressions

3. Solving Logarithmic Equations Using the Logarithm Property of Equality

4. Using the Change of Base Formula

Objective 1: Using the Product Rule, Quotient Rule and Power Rule for Logarithms

In this section we will learn how to manipulate logartihmic expressions using properties of logarithms. Understanding how to use these properties will help us solve exponential and logarithmic equations that we will encounter in the next section.

Recall from Section 5.3 the General Properties and Cancellation Properties

of Logarithms. We now look at three additional properties of logarithms.

Properties of Logarithms

If [pic], [pic]and v represent positive numbers, and r is any real number, then

[pic] Product Rule for Logarithms

[pic] Quotient Rule for Logarithms

[pic] Power Rule for Logarithms

Warning!

[pic].

[pic].

[pic].

[pic].

Objective 2: Expanding and Condensing Logarithmic Expressions

Sometimes it is necessary to combine several properties of logarithms to expand a logarithmic expression into the sum and/or difference of logarithms or to condense several logarithms into a single logarithm.

Objective 3: Solving Logarithmic Equations Using the Logarithm Property of Equality

Remember, all logarithmic functions of the form [pic] for [pic]are one-to-one. In

Section 3.6, the alternate definition of one-to-one stated that:

A function [pic]is one-to-one if for any two range values[pic]and[pic], [pic]

implies that [pic].

Using this definition and letting [pic], we can say that if [pic] , then [pic]. In other words, if the bases of a logarithmic equation of the form [pic] are equal, then the arguments must be

equal. This is known as the logarithm property of equality.

The Logarithm Property of Equality

If a logarithmic equation can be written in the form [pic], then [pic].

Furthermore, if [pic], then [pic].

The second statement of the logarithm property of equality says that if we start with an equation [pic], then we

can rewrite the equation as [pic]. This process is often casually referred to as “taking the log of both sides.”

Objective 4: Using the Change of Base Formula

Most scientific calculators are equipped with a [pic]key and a [pic] key to evaluate common logarithms and natural logarithms. We can also use a calculator to evaluate logarithmic expressions having a base other than 10 or e using the change of base formula.

Change of Base Formula

For any positive base [pic]and for any positive real number u, then

[pic]

where [pic]is any positive number such that [pic].

The change of base formula allows us to change the base of a logarithmic expression into a ratio of two logarithms using any base we choose. [pic]

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