1. Properties of Logarithms
Section 3.3
Properties of Logarithms

2. Objective By following instructions students will be able to:
Rewrite logarithmic functions with different bases.
Use properties of logarithms to evaluate or rewrite logarithmic expressions.
Use properties of logarithms to expand or condense logarithmic expressions.
Use logarithmic functions to model and solve real-life problems.

3. Change-of-Base Formula
Let a, b, and x be positive real numbers such that
and Then can be converted to a different base using any of the following formulas.
Base b
Base 10
Base e

4. Example 1:
Change bases using common logarithms. a) b)

5. Example 2:
Change bases using natural logarithms. a) b)

6. U-TRY #1
Rewrite the logarithm as a multiple of (a) a common logarithm and (b) a natural logarithm. 1) 2) 3) 4)

7. Properties of Logarithms
Let a be positive real numbers such that , and let n be a real number. If u and v are positive real numbers, the following properties are true. 1) 2) 3)

8. Example 3:
Write the logarithms in terms of ln2 and ln3. a) b)

9. Example 4:
Write the logarithms to verify that

10. Example 5:
Use the properties of logarithms to expand

11. Example 6:
Use the properties of logarithms to expand

12. U-TRY #2
Use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 1) 2) 3) 4)

13. Example 7:
Use the properties of logarithms to condense each logarithmic expression. a) b)

14. U-TRY #3 Write the expression as the logarithm of a single quantity.1) 2) 3) 4)

15. Example 8:
The table shows the mean distance from the sun x and the orbital period y of the six planets that are closer to the sun. In the table, the mean distance is given in terms of astronomical units (where the earth’s mean distance is defined as 1.0), and the period is given in terms of years. Find an equation that expresses y as a function of x.
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Mean Distance , x
0.387
0.723
1.0
1.524
5.203
9.539
Period, y
0.241
0.615
1.881
11.862
29.458

16. Revisit Objective Did we…
Rewrite logarithmic functions with different bases?
Use properties of logarithms to evaluate or rewrite logarithmic expressions?
Use properties of logarithms to expand or condense logarithmic expressions?
Use logarithmic functions to model and solve real-life problems?

17. Homework
Pg 244#s 1-93 EOO