1. 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
8 5𝑥 = 64 5𝑥−5
𝑙𝑜𝑔 7 49

2. Solving Logarithmic Equations
Lessons
Solving Logarithmic Equations
(log on one side)
(log on both sides)
Product Rule
Quotient Rule
Power Rule
Solving Logs

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

4. 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.

5. 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

6. 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

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

8. 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”

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

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

11. 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

12. 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)

13. 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

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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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.

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

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

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

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