1. 7-4 Properties of Logarithms Warm Up Lesson Presentation Lesson Quiz
Holt Algebra 2

2. Warm Up Simplify. 1. (26)(28) 214 2. (3–2)(35) 33 3. 38 4. 44 5. (73)5
715
Write in exponential form.
6. logx x = 1
7. 0 = logx1
x1 = x
x0 = 1

3. Objectives Use properties to simplify logarithmic expressions.
Translate between logarithms in any base.

4. Joke for ya 
Teacher: What are some properties of logs?
Student: They’re round, they have bark, and they come from trees. 

5. The logarithmic function for pH that you saw in the previous lessons, pH =–log[H+], can also be expressed in exponential form, as 10–pH = [H+].
Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents

6. Recall: What is the product of powers property?

7. The property in the previous slide can be used in reverse to write a sum of logarithms (exponents) as a single logarithm, which can often be simplified.
Helpful Hint
Think: logj + loga + logm = logjam

8. Example 1: Adding Logarithms
Express log64 + log69 as a single logarithm. Simplify.
log64 + log69
To add the logarithms, multiply the numbers.
log6 (4  9)
log6 36
Simplify. 2
Think: 6? = 36.

9. Check It Out! Example 1a
Express as a single logarithm. Simplify, if possible.
log log525
log5 (625 • 25)
To add the logarithms, multiply the numbers.
log5 15,625
Simplify. 6
Think: 5? = 15625

10. Express as a single logarithm. Simplify, if possible.
Check It Out! Example 1b
Express as a single logarithm. Simplify, if possible.
log log 1 3 9 1 3
log (27 • ) 9
To add the logarithms, multiply the numbers. 1 3
log 3
Simplify. –1
Think: ? = 3 1 3

11. Ok….so exponents….logarithms….inverses…..
BUT WHY!
Watch me 

12. Recall: Quotient of Powers Property
Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base.

13. The property above can also be used in reverse.
Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified.
Caution

14. Example 2: Subtracting Logarithms
Express log5100 – log54 as a single logarithm. Simplify, if possible.
log5100 – log54
To subtract the logarithms, divide the numbers.
log5(100 ÷ 4)
log525
Simplify. 2
Think: 5? = 25.

15. Check It Out! Example 2
Express log749 – log77 as a single logarithm. Simplify, if possible.
log749 – log77
To subtract the logarithms, divide the numbers
log7(49 ÷ 7)
log77
Simplify. 1
Think: 7? = 7.

16. BUT WHY! 

17. CAUTION:
log(ab) IS NOT the same as log(a)*log(b)
Log(a/b) IS NOT the same as log(a)/log(b)

18. Because you can multiply logarithms, you can also take powers of logarithms.

19. Example 3: Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
A. log2326
B. log8420
6log232
20log84
Because = 32, log232 = 5.
Because = 4, log84 = 2 3
6(5) = 30
20( ) = 40 3 2

20. Check It Out! Example 3
Express as a product. Simplify, if possibly.
a. log104
b. log5252
4log10
2log525
Because = 10, log 10 = 1.
Because = 25, log525 = 2.
4(1) = 4
2(2) = 4

21. Express as a product. Simplify, if possibly.
Check It Out! Example 3
Express as a product. Simplify, if possibly.
c. log2 ( )5 1 2
5log2 ( ) 1 2
Because 2–1 = , log2 = –1. 1 2
5(–1) = –5

22. Lesson Quiz:
Express each as a single logarithm.
1. log69 + log624
log6216 = 3
2. log3108 – log34
log327 = 3
Simplify.
3. log2810,000
30,000
4. log44x –1
x – 1