1. 6.5 Properties of Logarithms
MAT SPRING 2009
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

2. Properties of Logarithmic Functions
MAT SPRING 2009
Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we can make use of the following properties of logarithms.
Properties of Logarithms

3. Properties of Logarithms
The four properties of logarithms listed earlier hold true for natural logarithms as well.
Properties of Natural Logarithms

4. Examples
Find the exact value of each expression without using a calculator.

5. Other Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms Natural Logarithms

6. Properties of Logarithms
WARNING!!!!!!

7. Properties of Logarithms
ANOTHER WARNING!!!!!!

8. 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).

9. Using the Properties of Logarithms to Expand Log Expressions

10. Using the Properties of Logarithms to Expand Log Expressions

11. Using the Properties of Logarithms to Expand Log Expressions
MAT SPRING 2009
Using the Properties of Logarithms to Expand Log Expressions

12. Using the Properties of Logarithms to Condense Log Expressions
Use the properties of logs to write each expressions as a single logarithm (CONDENSE).

13. Using the Properties of Logarithms to Condense Log Expressions

14. Using the Properties of Logarithms to Condense Log Expressions

15. 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.

16. 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.

17. Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then logax can be converted to a different base using any of the following formulas.

18. Change-of-Base Formula Examples
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.

19. Change-of-Base Formula Examples
Use the change-of-base formula to evaluate:
a) b)

20. MAT SPRING 2009
End of Section 6.5