1. Warm-Up:
Graph the logarithmic function and exponential function then state the domain and range for each.
D: x-int:
R: y-int:
HA:
D: x-int:
R: y-int:
VA:
HW pages (EVEN 12-32, 42-56, 70-88)

2. 3.3 Properties of Logarithms Objective:
Change bases of logarithms
Apply properties of logarithms
Rewrite logarithmic expressions by expansion and contraction

3. Recall: Properties of Logarithms
logaa = 1
logaax = x
If loga x = loga y
then x = y

4. Remember Common Logarithm??
Is the “common” logarithm
This is the LOG button on the calculator
A log is implied to be base ten when we don’t write the base…
log107 = log 7

5. Remember Natural Logarithm??
loge
Is the “natural” logarithm
This is the ln button on the calculator
ln is used to represent loge …
loge7 = ln 7

6. Product Property Express as a sum of logarithms. 1) log43N
= log43 + log4N
2) ln6
= ln2 + ln3
3) ln19 + ln3
= ln(193)
= ln (57)

7. Ex. Express as a sum of logarithms, then simplify 4) log2 (416)
= log2 4 + log216
= log log224
= 2 + 4
= 6

8. Quotient Property

9. Ex.
Express as the difference of logs 5) 6) 7)

10. Power Property

11. Ex.
Express as a product.
= -5 logb9 8) 9)

12. Rewrite the logarithm of a quotient
10)

13. Change of base Formula
Let a, b, and x be positive real numbers such that and b do not = 1. then log base a of x can be converted to a different base as follows.

14. Example Rewrite the logarithm as a ratio of (a) common logs and
(b) natural logs.

15. Example
Rewrite as a common log using change of base

16. Expand.
log105x3y
log105
+ log10x3
+ log10y
log105
+ 3 log10x
+ log10y

17. Expand Simplify the division. Simplify the multiplication of 4
Change the radical sign to an exponent
Express the exponent as a product

18. Ex.
Condense.

19. Condense Express all products as exponents
Change the fractional exponent to a radical sign.
Simplify the subtraction.
Simplify the addition.

20. Summary: Properties of Logarithms
because a0 = 1
logaa = 1
because a1 = a
logaax = x
If loga x = loga y
then x = y
Product Property
Quotient Property
Power Property
Change-of-Base

21. Sneedlegrit: Expand: Condense:
HW pages (EVEN 12-32, 42-56, 70-88)