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

2. Opener-NEW SHEET-12/9 Simplify. 1. (26)(28) 214 2. (3–2)(35) 33 3. 384. 44
5. (73)5
715
Write in exponential form.
6. logx x = 1
7. 0 = logx1
x1 = x
x0 = 1

3. 7-3 Hmwk Quiz
1. Change 64 = 1296 to logarithmic form.
2. Change log279 = to exponential form. 2 3
Calculate the following using mental math.
3. log 10,000
4. log264

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

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. 7-4 Explore

8. Example 1: Adding Logarithms
Express log64 + log69 as a single logarithm. Simplify.
log log525
log log 1 3 9

9. Example 2: Subtracting Logarithms
Express log5100 – log54 as a single logarithm. Simplify, if possible.
Express log749 – log77 as a single logarithm. Simplify, if possible.

10. Express as a product. Simplify, if possibly.
Check It Out! Example 3
Express as a product. Simplify, if possibly.
a. log104
b. log5252
c. log2 ( )5 1 2

11. Jigsaw

12. Example 4: Recognizing Inverses
Simplify each expression.
a. log3311
b. log381
c. 5log510
log3311
log33  3  3  3
5log510 11
log334 10 4

13. Check It Out! Example 5a Continued
Evaluate log816.
Evaluate log927.

15. Remember that to multiply powers with the same base, you add exponents.

16. 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

17. Remember that to divide powers with the same base, you subtract exponents
Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base.

18. 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

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

20. Example 3: Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
B. log8420

21. Exponential and logarithmic operations undo each other since they are inverse operations.

22. Check It Out! Example 4
a. Simplify log100.9

23. Most calculators calculate logarithms only in base 10 or base e (see Lesson 7-6). You can change a logarithm in one base to a logarithm in another base with the following formula.

24. Example 5 Continued
Evaluate log328.

25. Opener-Same sheet-12/12 2. Simplify 2log2(8x) 1. log5625 + log525
4. Evaluate log927.

27. Express each as a single logarithm. 1. log69 + log624 log6216 = 3
Team Problems
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
5. 10log125
125
6. log64128 7 6

28. Signs

29. Type 2
Pick any of the two properties and describe how to solve them and give an example of how to do them.

30. Use a calculator to find each logarithm to the nearest thousandth.
Lesson Quiz: Part II
Use a calculator to find each logarithm to the nearest thousandth.
7. log320
2.727
8. log 10 1 2
–3.322
9. How many times as much energy is released by a magnitude-8.5 earthquake as a magntitude-6.5 earthquake?
1000