Solve the equations. 3 3𝑥 = 27 2𝑥+12 16 −𝑥 = 8 2𝑥−10 Warm Up Solve the equations. 3 3𝑥 = 27 2𝑥+12 16 −𝑥 = 8 2𝑥−10

7.3 Logarithms Let’s begin this lesson with a puzzle. See if you can figure out the pattern and answer the following questions.

Let’s start with a SUPER FUN puzzle! Now see if you can fill in the blanks: 𝑝𝑜𝑤𝑒𝑟 2 16 = 𝑝𝑜𝑤𝑒𝑟 6 36 = 𝑝𝑜𝑤𝑒𝑟 5 =3 𝑝𝑜𝑤𝑒𝑟 2 =1 𝑝𝑜𝑤𝑒𝑟 10 1000 = 𝑝𝑜𝑤𝑒𝑟 81 =2 𝑝𝑜𝑤𝑒𝑟 81 =4 𝑝𝑜𝑤𝑒𝑟 16 = 1 2 Take a guess at what these statements are saying: 𝑝𝑜𝑤𝑒𝑟 2 8 =3 𝑝𝑜𝑤𝑒𝑟 2 32 =5 𝑝𝑜𝑤𝑒𝑟 3 9 =2 𝑝𝑜𝑤𝑒𝑟 3 81 =4 𝑝𝑜𝑤𝑒𝑟 5 25 =2

Logarithm Overview What power of 6 equals 36? Power,6 36=2 log,6 36=2 6^2=36

A logarithm is an exponent (a power). 𝑙𝑜𝑔 2 32 =5 𝑙𝑜𝑔 3 27 =3 Notation A logarithm is an exponent (a power). 𝑙𝑜𝑔 2 32 =5 𝑙𝑜𝑔 3 27 =3 Say it: ____________________ What it means: Say it: ____________________ What it means:

Inverses What are some familiar examples of inverses? Exponentials also have an inverse function: ____________ The inverse of 10 2.6 =398 is ________ Def: 𝑙𝑜𝑔 𝑏 𝐴=𝑥 is the inverse of 𝑏 𝑥 =𝐴

Switching Forms Change between exponential and logarithmic equations… 𝟗 𝟑 =𝟕𝟐𝟗 𝒍𝒐𝒈 𝟗 𝟕𝟐𝟗 =𝟑

↔ Switching Forms 𝑏 𝑥 =𝐴 𝑙𝑜𝑔 𝑏 𝐴=𝑥 𝑙𝑜𝑔 6 216=3 49 2 =7 𝑏 𝑥 =𝐴 𝑙𝑜𝑔 𝑏 𝐴=𝑥 “exponential form” “log form”

Example Write each equation in exponential form. 𝑙𝑜𝑔 4 16=2 𝑙𝑜𝑔 3 729=6 𝑙𝑜𝑔 2 64=6

You Try! Write each equation in exponential form. 𝑙𝑜𝑔 3 9=2 𝑙𝑜𝑔 10 1,000=3 𝑙𝑜𝑔 5 625=4

Write each equation in logarithmic form. 15 3 =3375 4 1 2 =2 4 3 =64 Example Write each equation in logarithmic form. 15 3 =3375 4 1 2 =2 4 3 =64

Write each equation in logarithmic form. 3 3 =27 125 1 3 =5 6 2 =36 You Try! Write each equation in logarithmic form. 3 3 =27 125 1 3 =5 6 2 =36

Summary Convert each logarithmic expression to an exponential, and convert each exponential to a logarithmic. 4 3 =64 𝑙𝑜𝑔 12 12=1 𝑙𝑜𝑔 5 25=2 6 −2 = 1 36

Solving Logs To evaluate (solve) logarithms the first step is to convert your problem to an exponential! 𝑙𝑜𝑔 2 16

Logarithm Loop 𝑙𝑜𝑔 2 16=𝑥

Evaluate each expression. 𝑙𝑜𝑔 3 81 𝑙𝑜𝑔 4 8 𝑙𝑜𝑔 2 4 Examples Evaluate each expression. 𝑙𝑜𝑔 3 81 𝑙𝑜𝑔 4 8  𝑙𝑜𝑔 2 4

Evaluate each expression. 𝑙𝑜𝑔 64 4 𝑙𝑜𝑔 4 4 𝑙𝑜𝑔 125 5 Examples Evaluate each expression. 𝑙𝑜𝑔 64 4 𝑙𝑜𝑔 4 4  𝑙𝑜𝑔 125 5

Evaluate each expression. 𝑙𝑜𝑔 8 8 𝑙𝑜𝑔 5 25 𝑙𝑜𝑔 64 2 You Try! Evaluate each expression. 𝑙𝑜𝑔 8 8   𝑙𝑜𝑔 5 25 𝑙𝑜𝑔 64 2

HOMEWORK Lesson 7.3 Pg. 472 #s 13-15, 21-23, 27-30