Warm – up # 2 1. Rewrite in log form 5 4 =625 𝑙𝑜𝑔 5 625=4 2. Evaluate 𝑙𝑜𝑔 𝑥 128=7 𝑥 7 =128 𝑥 7 = 2 7 x = 2

Homework Log Tues 2/9 Lesson 7 – 4 Learning Objective: To rewrite logs using properties of logs Hw: Pg. 466 #9 – 37 odd

2/9/16 Lesson 7 – 4 Properties of Logs Day 1 Algebra II

Learning Objective To expand and condense logs using properties of logs

Properties of Logs Product Property 𝑙𝑜𝑔 𝑏 𝑚𝑛= 𝑙𝑜𝑔 𝑏 𝑚+ 𝑙𝑜𝑔 𝑏 𝑛 Quotient Property 𝑙𝑜𝑔 𝑏 𝑚 𝑛 = 𝑙𝑜𝑔 𝑏 𝑚− 𝑙𝑜𝑔 𝑏 𝑛 Power Property 𝑙𝑜𝑔 𝑏 𝑚 𝑛 =𝑛 𝑙𝑜𝑔 𝑏 𝑚

Expand the logs 1. 𝑙𝑜𝑔 3 9 𝑥 5 𝑦 = 𝑙𝑜𝑔 3 9+ 𝑙𝑜𝑔 3 𝑥 5 + 𝑙𝑜𝑔 3 𝑦 = 𝑙𝑜𝑔 3 9+ 5𝑙𝑜𝑔 3 𝑥+ 𝑙𝑜𝑔 3 𝑦 =2+ 5𝑙𝑜𝑔 3 𝑥+ 𝑙𝑜𝑔 3 𝑦 2. 𝑙𝑜𝑔 4𝑥 𝑦 = log 4x – log y = log 4 + log x – log y

Expand the logs 3. log ( 𝑚 2 𝑛 5 ) = log 𝑚 2 + log 𝑛 5 =2 log 𝑚 +5 log 𝑛 4. 𝑙𝑜𝑔 2 𝑥 = 𝑙𝑜𝑔 2 𝑥 1/2 = 1 2 𝑙𝑜𝑔 2 𝑥 or = 𝑙𝑜𝑔 2 𝑥 2

Expand the logs 5. 𝑙𝑜𝑔 4 𝑥 5 𝑦 7 𝑧 𝑤 4 = 𝑙𝑜𝑔 4 ( 𝑥 5 𝑦 7 ) 1/2 𝑧 𝑤 4 = 𝑙𝑜𝑔 4 𝑥 5/2 𝑦 7/2 𝑧 𝑤 4 = 𝑙𝑜𝑔 4 ( 𝑥 5/2 𝑦 7/2 )− 𝑙𝑜𝑔 4 (𝑧 𝑤 4 ) = 𝑙𝑜𝑔 4 𝑥 5 2 + 𝑙𝑜𝑔 4 𝑦 7 2 − 𝑙𝑜𝑔 4 𝑧− 𝑙𝑜𝑔 4 𝑤 4 = 5 2 𝑙𝑜𝑔 4 𝑥+ 7 2 𝑙𝑜𝑔 4 𝑦− 𝑙𝑜𝑔 4 𝑧−4 𝑙𝑜𝑔 4 𝑤

Expand the logs 6. 𝑙𝑜𝑔 4𝑥 𝑦 −3 𝑤 4 = 𝑙𝑜𝑔 ( 4𝑥 𝑦 −3 𝑤 4 ) 1/2 = 𝑙𝑜𝑔 4 1/2 𝑥 1/2 𝑦 −3/2 𝑤 2 = 𝑙𝑜𝑔( 4 1/2 𝑥 1/2 )−𝑙𝑜𝑔( 𝑦 −3/2 𝑤 2 ) = 𝑙𝑜𝑔2+𝑙𝑜𝑔 𝑥 1 2 −𝑙𝑜𝑔 𝑦 −3/2 −𝑙𝑜𝑔 𝑤 2 = 𝑙𝑜𝑔2+ 1 2 𝑙𝑜𝑔𝑥+ 3 2 𝑙𝑜𝑔𝑦−2𝑙𝑜𝑔𝑤

Condense/Write as a single log 7. 6 𝑙𝑜𝑔 5 𝑥+5 𝑙𝑜𝑔 5 𝑦− 𝑙𝑜𝑔 5 𝑧 = 𝑙𝑜𝑔 5 𝑥 6 + 𝑙𝑜𝑔 5 𝑦 5 − 𝑙𝑜𝑔 5 𝑧 = 𝑙𝑜𝑔 5 ( 𝑥 6 𝑦 5 𝑧 ) NOT 𝑙𝑜𝑔 5 ( 𝑥 6 + 𝑦 5 𝑧 ) NOT 𝑙𝑜𝑔 5 (𝑥 6 𝑦 5 ) 𝑙𝑜𝑔 5 𝑧 NOT 𝑙𝑜𝑔 5 (𝑥 6 𝑦 5 ) 𝑧

Condense/Write as a single log 8. 3 𝑙𝑜𝑔 2 𝑥+ 𝑙𝑜𝑔 2 5−2 𝑙𝑜𝑔 2 𝑧− 𝑙𝑜𝑔 2 7 = 𝑙𝑜𝑔 2 𝑥 3 + 𝑙𝑜𝑔 2 5− 𝑙𝑜𝑔 2 𝑧 2 − 𝑙𝑜𝑔 2 7 = 𝑙𝑜𝑔 2 ( 5 𝑥 3 7 𝑧 2 ) NOT 𝑙𝑜𝑔 2 ( 5+ 𝑥 3 𝑧 2 −7 ) NOT 𝑙𝑜𝑔 2 (5𝑥 3 ) 𝑙𝑜𝑔 2 (7 𝑧 2 ) NOT 𝑙𝑜𝑔 1 (5𝑥 2 ) 7 𝑧 2

Condense/Write as a single log 9. 2 𝑙𝑜𝑔 𝑏 𝑥 3 + 3 𝑙𝑜𝑔 𝑏 𝑦 4 −5 𝑙𝑜𝑔 𝑏 𝑧 = 𝑙𝑜𝑔 𝑏 𝑥 2/3 + 𝑙𝑜𝑔 𝑏 𝑦 3/4 − 𝑙𝑜𝑔 𝑏 𝑧 5 = 𝑙𝑜𝑔 𝑏 𝑥 2/3 𝑦 3/4 𝑧 5 Or = 𝑙𝑜𝑔 𝑏 3 𝑥 2 ∙ 4 𝑦 3 𝑧 5

Condense/Write as a single log 10. 𝑙𝑜𝑔 4 32− 𝑙𝑜𝑔 4 2 = 𝑙𝑜𝑔 4 32 2 = 𝑙𝑜𝑔 4 16 4 𝑥 =16 = 2 11. 𝑙𝑜𝑔 6 24− 𝑙𝑜𝑔 6 4 = 𝑙𝑜𝑔 6 24 4 = 𝑙𝑜𝑔 6 6 6 𝑥 =6 = 1

Condense/Write as a single log 12. 𝑙𝑜𝑔 64 2+ 𝑙𝑜𝑔 64 8 = 𝑙𝑜𝑔 64 (2∙8) = 𝑙𝑜𝑔 64 16 64 𝑥 =16 ( 2 6 ) 𝑥 = 2 4 2 6𝑥 = 2 4 6x = 4 𝑥= 2 3

Ticket Out the Door Condense into one log 1 2 𝑙𝑜𝑔 𝑥 4+ 𝑙𝑜𝑔 𝑥 𝑦 −3 𝑙𝑜𝑔 𝑥 𝑧 Check your work by expanding your answer

Assignment: Pg. 466 #9 – 37 odd

Extra Credit for Winners! BINGO Extra Credit for Winners!

Rewrite in Log Form

Solve for x.

Evaluate

Evaluate

Rewrite in Exponential Form.

Rewrite in Log Form

Evaluate

Evaluate

Evaluate

Evaluate

Evaluate

Rewrite in Log Form

Evaluate

Evaluate

Solve for n.

Rewrite in Exponential Form.

Rewrite in Log Form

Evaluate

Solve for x

Rewrite in Exponential Form.

Solve for a.

Rewrite in Exponential Form

Solve for m.

Evaluate

Evaluate

Rewrite in Exponential Form

Evaluate

Rewrite in Log Form

Solve for x.

Rewrite in Log Form

Solve for x.

Rewrite in Exponential Form.

Rewrite in Log Form

Solve for x

Rewrite in Exponential Form.