Section 5.4Properties of Logarithms
Objectives
- Using the Product Rule, Quotient Rule and Power Rule for Logarithms
- Expanding and Condensing Logarithmic Expressions
- Solving Logarithmic Equations Using the Logarithm Property of Equality
- 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 , and vrepresent positive numbers, and r is any real number, then
Product Rule for Logarithms
Quotient Rule for Logarithms
Power Rule for Logarithms
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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 for are one-to-one. In
Section 3.6, the alternate definition of one-to-one stated that:
A function is one-to-one if for any two range valuesand,
implies that.
Using this definition and letting , we can say that if , then . In other words, if the bases of a logarithmic equation of the form are equal, then the arguments must be
equal. This is known as the logarithm property of equality.
The Logarithm Property of Equality
If alogarithmic equation can be written in the form, then .
Furthermore, if , then .
The second statement of the logarithm property of equality says that if we start with an equation, then we
can rewrite the equation as . 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 key and a 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 thechange of base formula.
Change of Base Formula
For any positive base and for any positive real number u, then
where is any positive number such that .
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.