Holt McDougal Algebra 2 7-4 Properties of Logarithms 7-4 Properties of Logarithms Holt Algebra 2...

Holt McDougal Algebra 2

7-4 Properties of Logarithms7-4 Properties of Logarithms

Holt Algebra 2

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7-4 Properties of Logarithms

Opener-NEW SHEET-12/9

2. (3–2)(35)214 33

38

1. (26)(28)

3. 4.

5. (73)5 715

44

Simplify.

Write in exponential form.

x0 = 16. logx x = 1 x1 = x 7. 0 = log

x1



7-3 Hmwk Quiz

1. Change 64 = 1296 to logarithmic form.

2. Change log279 = to exponential form.23

3. log 10,000

4. log264

Calculate the following using mental math.



Use properties to simplify logarithmic expressions.

Translate between logarithms in any base.

Objectives



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



7-4 Explore



Express log64 + log

69 as a single logarithm.

Simplify.

Example 1: Adding Logarithms

log5625 + log

525

log

27 + log13

13

19



Express log5100 – log


Simplify, if possible.

Example 2: Subtracting Logarithms

Express log749 – log


Simplify, if possible.



Express as a product. Simplify, if possibly.

a. log104 b. log5252

Check It Out! Example 3

c. log2 ( )5 1

2



Jigsaw



Example 4: Recognizing Inverses

Simplify each expression.

b. log381 c. 5log510a. log

3311

log3311

11

log33 3 3 3

log334

4

5log510

10



Evaluate log927.

Check It Out! Example 5a Continued

Evaluate log816.



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



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.

Think: logj + loga + logm = logjam

Helpful Hint



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.



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



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



Express as a product. Simplify, if possible.

Example 3: Simplifying Logarithms with Exponents

B. log8420



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



a. Simplify log100.9

Check It Out! Example 4



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.



Example 5 Continued

Evaluate log32

8.



Opener-Same sheet-12/12

1. log5625 + log

525 2. Simplify 2log

2(8x)

3. log2326

4. Evaluate log927.



Team ProblemsExpress each as a single logarithm.

1. log69 + log

624 log

6216 = 3

2. log3108 – log

34

Simplify.3. log

2810,000

log327 = 3

30,000

4. log44x –1 x – 1

6. log64

128 76

5. 10log125 125



Signs



Type 2• Pick any of the two properties and

describe how to solve them and give an example of how to do them.



Lesson Quiz: Part II

Use a calculator to find each logarithm to the nearest thousandth.

7. log320

9. How many times as much energy is released by a magnitude-8.5 earthquake as a magntitude-6.5 earthquake?

2.727

–3.322

1000

8. log 10 1

2

Holt McDougal Algebra 2 7-4 Properties of Logarithms 7-4 Properties of Logarithms Holt Algebra 2...

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