The WhatPower Function à An Introduction to Logarithms
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1 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
2 vi. WhatPower ;< = vii. WhatPower ;<< = viii. WhatPower > 64 = ix. WhatPower 4 64 = x. WhatPower? 3 = xi. 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 WhatPower < WhatPower ; 5 5. WhatPowerB H 9 S.5
3 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
4 d. 625 e. log ;< f. log ;<<< ( ) 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
5 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. = 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. 625 = 4 b. log ;< 0. = c. log 4F 9 = 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 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
6 f. log % 3 S.55
7 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. 325 b. (25) c. d. 25 e. (0) f 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
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