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

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

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

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

Recall: What is the product of powers property?

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

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.

Check It Out! Example 1a Express as a single logarithm. Simplify, if possible. log5625 + log525 log5 (625 • 25) To add the logarithms, multiply the numbers. log5 15,625 Simplify. 6 Think: 5? = 15625

Express as a single logarithm. Simplify, if possible. Check It Out! Example 1b Express as a single logarithm. Simplify, if possible. log 27 + 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

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

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.

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

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.

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.

BUT WHY! 

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

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

Example 3: Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. A. log2326 B. log8420 6log232 20log84 Because 25 = 32, log232 = 5. Because 8 = 4, log84 = . 2 3 6(5) = 30 20( ) = 40 3 2

Check It Out! Example 3 Express as a product. Simplify, if possibly. a. log104 b. log5252 4log10 2log525 Because 101 = 10, log 10 = 1. Because 52 = 25, log525 = 2. 4(1) = 4 2(2) = 4

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

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