Good afternoon Happy Thursday!!! Warm UP: Convert from logs to exponentials or vice versa 1) log 2 𝑥−2 =18 2) 𝑒 𝑥 =20 Solve by changing the bases: 2 𝑥+4 =32 1 9 𝑥−5 =27

Law of Logs

Review laws of exponents 𝑥 𝑛 ∗ 𝑥 𝑚 = 𝑥 𝑛+𝑚 𝑥 𝑛 𝑥 𝑚 = 𝑥 𝑛−𝑚 𝑥 𝑛 𝑚 = 𝑥 𝑛∗𝑚 𝑥𝑦 𝑛 = 𝑥 𝑛 ∗ 𝑦 𝑛 𝑎𝑛𝑑 𝑥 𝑦 𝑛 = 𝑥 𝑛 𝑦 𝑛 𝑥 −𝑛 = 1 𝑥 𝑛 and 1 𝑥 −𝑛 = 𝑥 𝑛

Law of Logs Just like with Exponent Laws(rules) Logs Rules can change the look of a logarithm

Multiplication log 𝑏 𝑀𝑁= log 𝑏 𝑀 + log 𝑏 𝑁 Similar to : Product Rule Ex: log 2 4𝑥 = log 2 4+ log 2 𝑥

Division log 𝑏 𝑀 𝑁 = log 𝑏 𝑀 − log 𝑏 𝑁 Similar to Quotient Rule Ex: log 2𝑥 5 = log 2𝑥 −𝑙𝑜𝑔5

Power Rule log 𝑏 𝑀 𝑥 =𝑥𝑙𝑜 𝑔 𝑏 𝑀 Ex: log (10 2 ) =2 log 10 Test this one to be sure:

Logical Rules 𝐿𝑜 𝑔 𝑏 𝑀=𝐿𝑜 𝑔 𝑏 𝑁 if and only if 𝑀=𝑁 log b 𝑏 =1

Try it Confirm the equations are true log 10 =1 log 1 =0 𝑙𝑛 20 = ln 20

Practice: Write each log in expanded form log 5 (𝑥𝑦) log 5 𝑥 𝑦 2

More Practice Write each as single log. log 𝑥+ log 𝑦− log 𝑧 = 2𝑙𝑛𝑥+𝑙𝑛𝑦= 2 log 𝑥+3 log 𝑦+2 log 𝑧 = 1 2 ln 𝑥 −( 1 3 ) ln 𝑦 =

Homework Pg 405 #’s 23-28 odd, Pg 411 #’s 1-8, 35-38