Classwork Work with your partner or group to solve each of the following equations for x. a. 2 # = 2 % b. 2 # = 2 c. 2 # = 6 d. 2 # 64 = 0 e. 2 # = 0 f. 2 %# = 64 Exploring the WhatPower Function with Special Writing a. Evaluate each expression. The first two have been completed for you. i. (Example) WhatPower 4 8 = 3 ii. (Example) WhatPower % 9 = 2 iii. WhatPower 8 (36) = iv. WhatPower 4 (32) = v. WhatPower ;< 000 = S.50
vi. WhatPower ;< 000 000 = vii. WhatPower ;<< 000 000 = viii. WhatPower > 64 = ix. WhatPower 4 64 = x. WhatPower? 3 = xi. WhatPower @ 5 = xii. WhatPowerB C 8 = xiii. WhatPower >4 = b. With your group members, write a definition for the function WhatPower D, where b is a number. Exercises 5 Evaluate the following expressions, and justify your answers. If it s not possible, explain why it doesn t work.. WhatPower F 49 2. WhatPower < 7 3. WhatPower @ 4. WhatPower ; 5 5. WhatPowerB H 9 S.5
Making it More Formal with Real Math Words. log 4 8 = 3 2. log % 9 = 2 3. log 8 (36) = 4. log 4 (32) = 5. log ;< 000 = 6. log >4 = 7. log ;<< 0.0 = 8. log 4 4 = Exercise 0 0. Compute the value of each logarithm. Verify your answers using an exponential statement. a. log 4 32 b. log % 8 c. log? 8 S.52
d. log @ 625 e. log ;< 000000000 f. log ;<<< (000000000) g. log ;% 3 h. log ;% () i. log F 7 j. log? 27 k. log F 7 l. log F 49 m. log # (x 4 ) S.53
Lesson Summary If three numbers L, b, and x are related by x = b Q, then L is the logarithm base b of x, and we write log D (x) = L. That is, the value of the expression log D (x) is the power of b needed to obtain x. Valid values of b as a base for a logarithm are 0 < b < and b >. Problem Set. Rewrite each of the following in the form WhatPower D x = L. a. 3 @ = 243 b. 6 M% = 26 c. 9 < = 2. Rewrite each of the following in the form log D x = L. a. 6 B N = 2 b. 0 % = 000 c. b O = r 3. Rewrite each of the following in the form b Q = x. a. log @ 625 = 4 b. log ;< 0. = c. log 4F 9 = 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 4 (024) b. log 4 (28) c. log 4 8 d. log 4 6 e. log 4 (0) f. log 4 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 % d. log % 3 e. log % (0) S.54
f. log % 3 S.55
6. 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 @ 325 b. log @ (25) c. log @ d. log @ 25 e. log @ (0) f. log @ 25 7. Is there any positive number b so that the expression log D (0) makes sense? Explain how you know. 8. Is there any positive number b so that the expression log D ( ) makes sense? Explain how you know. 9. Verify each of the following by evaluating the logarithms. a. log 4 (8) + log 4 (4) = log 4 (32) b. log % (9) + log % (9) = log % (8) c. log > (4) + log > (6) = log > (64) d. log ;< (0 % ) + log ;< (0 > ) = log ;< (0 F ) 0. 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 D (x) + log D (y)?. To evaluate log 4 (3), Autumn reasoned that since log 4 2 = and log 4 4 = 2, log 4 (3) must be the average of and 2 and therefore log 4 3 =.5. Use the definition of logarithm to show that log 4 (3) cannot be.5. Why is her thinking not valid? 2. Find the value of each of the following. a. If x = log 4 (8) and y = 2 #, find the value of y. b. If log 4 (x) = 6, find the value of x. c. If r = 2 8 and s = log 4 (r), find the value of s. S.56