Solve the equations. 4 2𝑥 = 8 𝑥+7 𝑙𝑜𝑔 125 25 8 5𝑥 = 64 5𝑥−5 𝑙𝑜𝑔 7 49 Warm Up Solve the equations. 4 2𝑥 = 8 𝑥+7 𝑙𝑜𝑔 125 25 8 5𝑥 = 64 5𝑥−5 𝑙𝑜𝑔 7 49

Solving Logarithmic Equations Lessons 7.4-7.5 Solving Logarithmic Equations (log on one side) (log on both sides) Product Rule Quotient Rule Power Rule Solving Logs

Solving Logarithm Equations with a logarithm on one side and a number on the other 𝒍𝒐𝒈 𝟒 𝒙= 𝟓 𝟐

7.4 Logarithm Equations Let’s begin this lesson with a puzzle. See if you can figure out a way to solve the following questions based on the previous lesson.

𝑙𝑜𝑔 9 𝑥= 3 2 7.4 Logarithm Equations Convert the logarithm equation into an exponential equation to solve NO MATTER where the 𝑥 is! 𝑙𝑜𝑔 9 𝑥= 3 2

Solve the following logarithmic equations. 𝑙𝑜𝑔 16 𝑥= 5 2 𝑙𝑜𝑔 81 𝑥= 3 4 Practice Solve the following logarithmic equations. 𝑙𝑜𝑔 16 𝑥= 5 2 𝑙𝑜𝑔 81 𝑥= 3 4

Solving Logarithm Equations with a logarithm both sides 𝒍𝒐𝒈 𝟓 𝒙+ 𝒍𝒐𝒈 𝟓 𝟑= 𝒍𝒐𝒈 𝟓 𝟔

The bases must be the same! 7.5 Properties Where have we talked about the “Property of Equality” before? If 𝑙𝑜𝑔 𝑏 𝑥= 𝑙𝑜𝑔 𝑏 𝑦, then 𝑥=𝑦. The bases must be the same! Example: 𝑙𝑜𝑔 10 𝑥= 𝑙𝑜𝑔 10 (5𝑥−20) “delete the logs”

Practice Solve the following logarithmic equations. 𝑙𝑜𝑔 4 𝑥 2 = 𝑙𝑜𝑔 4 (−6𝑥−8) 𝑙𝑜𝑔 3 𝑥 2 −15 = 𝑙𝑜𝑔 3 2𝑥

You Try!!! Solve the following logarithmic equations. 𝑙𝑜𝑔 2 𝑥−4 = 𝑙𝑜𝑔 2 3𝑥 𝑙𝑜𝑔 5 𝑥 2 −10 = 𝑙𝑜𝑔 5 3𝑥

Product Property of Logarithms 𝑙𝑜𝑔 𝑏 𝑥𝑦 = 𝑙𝑜𝑔 𝑏 𝑥+ 𝑙𝑜𝑔 𝑏 𝑦 Logarithm of a Product Product Property of Logarithms 𝑙𝑜𝑔 𝑏 𝑥𝑦 = 𝑙𝑜𝑔 𝑏 𝑥+ 𝑙𝑜𝑔 𝑏 𝑦

Express as a sum of logarithms. 𝑙𝑜𝑔 𝑎 16∗32 𝑙𝑜𝑔 𝑏 (8∗16) 𝑙𝑜𝑔 𝑧 (3𝑎𝑏) Practice “the log of a product is the sum of the logs” Express as a sum of logarithms. 𝑙𝑜𝑔 𝑎 16∗32 𝑙𝑜𝑔 𝑏 (8∗16) 𝑙𝑜𝑔 𝑧 (3𝑎𝑏)

Practice Express as a single logarithms. 𝑙𝑜𝑔 𝑦 65+ 𝑙𝑜𝑔 𝑦 2 “the sum of the logs is the log of the products” Express as a single logarithms. 𝑙𝑜𝑔 𝑦 65+ 𝑙𝑜𝑔 𝑦 2 𝑙𝑜𝑔 𝑏 𝐻+ 𝑙𝑜𝑔 𝑏 K 𝑙𝑜𝑔 𝑘 8+ 𝑙𝑜𝑔 𝑘 𝑎

Logarithm of a Quotient Quotient Property of Logarithms 𝑙𝑜𝑔 𝑏 𝑥 𝑦 = 𝑙𝑜𝑔 𝑏 𝑥− 𝑙𝑜𝑔 𝑏 𝑦

Express as a difference of logarithms. 𝑙𝑜𝑔 𝑘 6 5 𝑙𝑜𝑔 𝑏 𝑦 𝑥 𝑙𝑜𝑔 𝑧 8 3 Practice “the log of a quotient is the difference of the logs” Express as a difference of logarithms. 𝑙𝑜𝑔 𝑘 6 5 𝑙𝑜𝑔 𝑏 𝑦 𝑥 𝑙𝑜𝑔 𝑧 8 3

Practice Express as a single logarithm. 𝑙𝑜𝑔 𝑘 42− 𝑙𝑜𝑔 𝑘 7 “the difference of the logs is the log of the quotient” Express as a single logarithm. 𝑙𝑜𝑔 𝑘 42− 𝑙𝑜𝑔 𝑘 7 𝑙𝑜𝑔 𝑏 𝐴− 𝑙𝑜𝑔 𝑏 𝐶 𝑙𝑜𝑔 𝑡 5− 𝑙𝑜𝑔 𝑡 13

Power Property of Logarithms 𝑙𝑜𝑔 𝑏 𝑥 𝑎 = 𝑎∗𝑙𝑜𝑔 𝑏 𝑥 Logarithm of a Power Power Property of Logarithms 𝑙𝑜𝑔 𝑏 𝑥 𝑎 = 𝑎∗𝑙𝑜𝑔 𝑏 𝑥

Express as a product. 𝑙𝑜𝑔 𝑏 𝑡 2 𝑙𝑜𝑔 3 𝑀 −2 Practice Express as a product. 𝑙𝑜𝑔 𝑏 𝑡 2 𝑙𝑜𝑔 3 𝑀 −2

Express as a single logarithm. 5 𝑙𝑜𝑔 𝑏 2 −2𝑙𝑜𝑔 3 10 Practice Express as a single logarithm. 5 𝑙𝑜𝑔 𝑏 2 −2𝑙𝑜𝑔 3 10

Solving Logarithms Use a property of logarithms to combine the left side of the equation. Use the equality property of logarithms to write a new equation. Solve the equation for 𝑥.

Practice Solve each equation. 𝑙𝑜𝑔 6 𝑥+ 𝑙𝑜𝑔 6 9= 𝑙𝑜𝑔 6 54 𝑙𝑜𝑔 9 3𝑥+14 − 𝑙𝑜𝑔 9 5= 𝑙𝑜𝑔 9 2𝑥

Practice Solve each equation. 4𝑙𝑜𝑔 2 𝑥+ 𝑙𝑜𝑔 2 5= 𝑙𝑜𝑔 2 405 𝑙𝑜𝑔 3 𝑦=− 𝑙𝑜𝑔 3 16+ 1 3 𝑙𝑜𝑔 3 64

YOU TRY!!! 𝑙𝑜𝑔 8 48− 𝑙𝑜𝑔 8 𝑤= 𝑙𝑜𝑔 8 4 Solve each equation. 𝑙𝑜𝑔 2 𝑥=5 𝑙𝑜𝑔 2 2− 𝑙𝑜𝑔 2 8

HOMEWORK Lesson 7.4-7.5 Pg. 480 #’s 11-15 Pg. 488 #’s 23-26