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6.5 Properties of Logarithms. In this section, we will study the following topics: Using the properties of logarithms to evaluate log expressions Using the properties of logarithms to expand or condense log expressions Using the change-of-base formula. Properties of Logarithmic Functions.

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6.5 Properties of Logarithms


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6 5 properties of logarithms
6.5 Properties of Logarithms

In this section, we will study the following topics:

Using the properties of logarithms to evaluate log expressions

Using the properties of logarithms to expand or condense log expressions

Using the change-of-base formula

slide2

Properties of Logarithmic Functions

To solve various types of problems involving logarithms, we can make use of the following properties of logarithms.

slide3

Properties of Logarithms

The four properties of logarithms listed earlier hold true for natural logarithms as well.

examples
Examples
  • Find the exact value of each expression without using a calculator.
slide7

Properties of Logarithms

ANOTHER WARNING!!!!!!

slide8

Using the Properties of Logarithms to Expand Log Expressions

Use the properties of logs to write the expression as the sum and/or difference of logarithms (EXPAND).

slide12

Using the Properties of Logarithms to Condense Log Expressions

Use the properties of logs to write each expressions as a single logarithm (CONDENSE).

slide15

Change-of-Base Formula

  • I mentioned in a previous section that the calculator only has two types of log keys:
    • COMMON LOG (BASE 10)
    • NATURAL LOG (BASE e).
  • It’s true that these two types of logarithms are used most often, but sometimes we want to evaluate logarithms with bases other than 10 or e.
slide16

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF BASE FORMULA.

We will convert the logarithm with base a into an equivalent expression involving common logarithms or natural logarithms.

slide18

Change-of-Base Formula Examples

Example*:

Use the change-of-base formula to evaluate log7264

using common logarithms

using natural logarithms.

Solution:

The result is the same whether you use the common log or the natural log.

slide19

Change-of-Base Formula Examples

Example:

Use the change-of-base formula to evaluate:

a) b)