NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M3 ALGEBRA II Lesson 8: The “WhatPower” Function Classwork Opening Exercise a.
Evaluate each expression. The first two have been completed for you. i.
WhatPower! 8 = 3 This is like saying “2 to what power is 8?” ii.
WhatPower! 9 = 2 This is like saying “3 to what power is 9?” iii.
WhatPower! (36) = ______ This is like saying “6 to what power is 36?” iv.
WhatPower! (32) =______ v.
WhatPower!" 1000 = ______ vi.
WhatPower!" 1000 000 =______ vii. WhatPower!"" 1000 000 =______ viii. WhatPower! 64 =______ ix.
WhatPower! 64 = ______ x.
WhatPower! 3 = ______ xi.
5 = ______ WhatPower!
xii. WhatPower!
!
1
= ______ 8
xiii. WhatPower!" 1 =______ xiv. WhatPower!"" 0.01 =______ xv. WhatPower!
1
= ______ 4
Lesson 8: The “WhatPower” Function This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-­‐math.org This file derived from ALG II-­‐M3-­‐TE-­‐1.3.0-­‐08.2015 S. 50 This work is licensed under a Creative Commons Attribution-­‐NonCommercial-­‐ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M3 ALGEBRA II xvi. WhatPower! 2 = ______ !
Exercises 1–9 Evaluate the following expressions, and justify your answers by writing in exponential form. 1.
WhatPower! 49 2.
WhatPower! 7 3.
WhatPower! 1 4.
WhatPower! 5 5.
WhatPower!! 16 6.
WhatPower!! 32 7.
WhatPower! 9 !
8.
WhatPower!! 27 !
Lesson 8: The “WhatPower” Function This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-­‐math.org This file derived from ALG II-­‐M3-­‐TE-­‐1.3.0-­‐08.2015 S. 51 This work is licensed under a Creative Commons Attribution-­‐NonCommercial-­‐ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M3 ALGEBRA II Examples We wil now do the same activity but we will change “whatpower” with “log” 1.
log ! 8 = 3 2.
log ! 9 = 2 3.
log ! (36) = ______ 4.
log ! (32) =______ 5.
log!" 1000 = ______ 6.
log !" 1 =______ 7.
log!"" 0.01 =______ 8.
log !
1
= ______ 4
Exercise 10 10. Compute the value of each logarithm. Verify your answers using an exponential statement. a.
log ! 32 b.
log ! 81 c.
log ! 81 d.
log ! 625 Lesson 8: The “WhatPower” Function This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-­‐math.org This file derived from ALG II-­‐M3-­‐TE-­‐1.3.0-­‐08.2015 S. 52 This work is licensed under a Creative Commons Attribution-­‐NonCommercial-­‐ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M3 ALGEBRA II e.
log!" 1000000000 f.
log!""" (1000000000) g.
log!" 13 h.
log!" (1) i.
log !
j.
log ! 27 k.
log
!
l.
log
!
7 7 1
49
m. log ! (𝑥 ! ) Lesson 8: The “WhatPower” Function This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-­‐math.org This file derived from ALG II-­‐M3-­‐TE-­‐1.3.0-­‐08.2015 S. 53 This work is licensed under a Creative Commons Attribution-­‐NonCommercial-­‐ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 8 M3 ALGEBRA II Lesson Summary §
If three numbers 𝐿, 𝑏, and 𝑥 are related by 𝑥 = 𝑏 ! , then 𝐿 is the logarithm base 𝑏 of 𝑥 , and we write log ! (𝑥) = 𝐿. That is, the value of the expression log ! (𝑥) is the power of 𝑏 needed to obtain 𝑥 . §
Valid values of 𝑏 as a base for a logarithm are 0 < 𝑏 < 1 and 𝑏 > 1. Problem Set 1.
Rewrite each of the following in the form WhatPower! 𝑥 = 𝐿. a.
3! = 243 b.
6!! =
1
216
c.
9! = 1 c.
𝑏 ! = 𝑟 c.
log !" 9 = 2.
Rewrite each of the following in the form log ! 𝑥 = 𝐿. a.
!
16! = 2 b.
10! = 1000 3.
Rewrite each of the following in the form 𝑏 ! = 𝑥. a.
log ! 625 = 4 b.
log!" 0.1 = −1 2
3
4.
Consider the logarithms base 2. For each logarithmic expression below, either calculate the value of the expression or explain why the expression does not make sense. a.
log ! (1024) b.
log ! (128) c.
log !
8 d.
log !
1
16
e.
log ! (0) f.
log ! −
1
32
5.
Consider the logarithms base 3. For each logarithmic expression below, either calculate the value of the expression or explain why the expression does not make sense. a.
log ! (243) b.
log ! (27) c.
log ! 1 d.
log !
e.
log ! (0) f.
log ! −
1
3
1
3
Lesson 8: The “WhatPower” Function This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-­‐math.org This file derived from ALG II-­‐M3-­‐TE-­‐1.3.0-­‐08.2015 S. 54 This work is licensed under a Creative Commons Attribution-­‐NonCommercial-­‐ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM 6.
Lesson 8 M3 ALGEBRA II Consider the logarithms base 5. For each logarithmic expression below, either calculate the value of the expression or explain why the expression does not make sense. a.
log ! 3125 b.
log ! (25) c.
log ! 1 d.
log !
e.
log ! (0) f.
log ! −
1
25
1
25
7.
Is there any positive number 𝑏 so that the expression log ! (0) makes sense? Explain how you know. 8.
Is there any positive number 𝑏 so that the expression log ! (−1) makes sense? Explain how you know. 9.
Verify each of the following by evaluating the logarithms. a.
log ! (8) + log ! (4) = log ! (32) b.
log ! (9) + log ! (9) = log ! (81) c.
log ! (4) + log ! (16) = log ! (64) d.
log!" (10! ) + log!" (10! ) = log!" (10! ) 10. Looking at the results from Problem 9, do you notice a trend or pattern? Can you make a general statement about the value of log ! (𝑥) + log ! (𝑦)? 11. To evaluate log ! (3), Autumn reasoned that since log ! 2 = 1 and log ! 4 = 2, log ! (3) must be the average of 1 and 2 and therefore log ! 3 = 1.5. Use the definition of logarithm to show that log ! (3) cannot be 1.5. Why is her thinking not valid? 12. Find the value of each of the following. a.
If 𝑥 = log ! (8) and 𝑦 = 2 ! , find the value of 𝑦. b.
If log ! (𝑥) = 6, find the value of 𝑥 . c.
If 𝑟 = 2! and 𝑠 = log ! (𝑟), find the value of 𝑠. Lesson 8: The “WhatPower” Function This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-­‐math.org This file derived from ALG II-­‐M3-­‐TE-­‐1.3.0-­‐08.2015 S. 55 This work is licensed under a Creative Commons Attribution-­‐NonCommercial-­‐ShareAlike 3.0 Unported License.