Algebra 2/Pre-Calculus Name__________________ Introduction to Logarithms (Days 1 and 2, Logarithmic Functions) In this problem set, we will introduce logarithms.

Definition:

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logb M = The answer to the question “b to what power equals M?” Examples:

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log2 8 = 3 because

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23 = 8

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log7 49 = 2 because

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72 = 49 ( ) 1log 5

15 −= because

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5−1 = 15

Definition:

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logM is the same as

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log10 M Example:

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log10000 = 4 because

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104 =10000

1. Evaluate each of the following logarithms without using your calculator.

a. 9log3 b. 16log2 c. 1000log

d. 101

log e. 7log7 f. 56 6log

g. 125log5 h. 35 5log i. 11

7 7log

j. 81

log2 k. 321log2 l.

31

log3

m. 1log7 n. 1log5 o. 3/210log Answers a. 2 b. 4 c. 3 d. -1 e. 21 f. 5 g. 3 h. 31 i. 11 j. -3 k. -5 l. 2

1− m. 0 n. 0 o. 32

2. Can you find the values of )100log(− ? )4(log2 − ? )0(log7 ? Explain.

3. In this problem, we will combine logarithms with other operations.

a. Evaluate 49log7 .

b. Evaluate 49log5 7 . Note: 49log5 7 means “5 times 49log7 ” in the same way that x5 means “5 times x.”

c. Evaluate 10)(log2 91

3 +− .

d. Evaluate ( )10000loglog2 .

e. Is )248(log2 + the same as 24)8(log2 + ? Find the value of both.

f. Is )84(log2 ⋅ the same as )8(log)4(log 22 ⋅ ? Find the value of both.

g. Is )8(log)4(log 22 + the same as )84(log2 + ? Find their values, if possible.

h. (Optional Challenge Problem) Look back at parts f and g of this problem. Notice anything unexpected? Make a conjecture and test it out. If possible, explain.

Some answers a. 2 b. 10 c. 14 d. 2 e. 5)248(log2 =+ , 2724)8(log2 =+ f. 5)84(log2 =⋅ , 6)8(log)4(log 22 =⋅ g. 5)8(log)4(log 22 =+ , )84(log2 + does not simplify

4. In this problem, we will learn how to switch between the “exponent form” of an equation and the “log form” of the equation.

a. Rewrite the equation 38log2 = as an equation without any logs.

b. The equation 38log2 = can be rewritten as 823 = . We could say that 38log2 = is the “log form” of the equation and 823 = is the “exponent form” of the equation. Rewrite the equation 8134 = in “log form.”

c. Rewrite the equation RPb =log in exponent form.

d. Rewrite the equation 5=xb in log form.

e. Rewrite the equation x=8log4 in exponent form.

Some answers b. 481log3 = c. PbR = d. xb =5log e. 84 =x

5. In this problem, we will find the value of 32log8 by switching forms.

a. Rewrite x=32log8 in exponent form.

b. You should have found that 328 =x . Now solve this equation for x. Hint: Rewrite all parts with a base of 2 and use exponent rules.

6. Here’s the solution for the last problem:

328 =x

53 2)2( =x

53 22 =x 53 =x

35=x

Thus, 35

8 32log = .

a. Confirm your answer by evaluating 3/58 on your calculator.

b. Show how you could evaluate 3/58 without using your calculator. Hint: Rewrite it as 53/1 )8( .

c. Use the technique from problem 5 to find the value of 2log4 .

Answer c. 41

4 2log =

7. Find the value of the following logarithms by converting to exponent form as you did in problems 5 and 6.

a. 27log9 b. ( )21

2log

c. 33log9 d. 100log 10

e. 4 1000log f. ⎟⎠

⎞⎜⎝

⎛37 77log

Answers a. 23 b. 21− c. 43 d. 4 e. 43 f. 32

8. Solve the following equation: 122)3(log5 5 =++x . Hint: Start with the easy steps. Then switch to exponent form.

Here’s the solution to the last problem:

122)3(log5 5 =++x

10)3(log5 5 =+x

2)3(log5 =+x

352 += x

325 += x

x=22

9. Find the value of 2)322(log5 5 ++ . How does this relate to the last problem?

10. Use the technique from problem 7 to solve each of the following equations. Note: You can always check your answer by plugging back into the original equation, as you did in problem 7.

a. 2log4 −=x b. 6)12(log3 7 =+x

c. 41)1log( =+−x d. 21

36 )24(log =−x

e. 31)(log4 8 =+x f. 83)7(log 22 =++x

g. 3)65(log 22 =−+ xx h. 21)log(2)log(9 += xx

Answers a. 161 b. 24 c. 1001 d. 2 e. 8 f. 5± g. -7, 2 h. 1000

11. Estimate the value of each of the following logarithms without using your calculator.

a. 25log3 b. 35log2 c. 68log d. )(log 101

3 We can evaluate logarithms on the calculator. The directions are slightly different depending on how new your calculator is. Here are the directions for finding 25log3 . New edition TI-84: Press the MATH button, scroll down to A: logBASE and press ENTER. Fill in the empty boxes so that you see 25log3 on the screen and press ENTER. Older edition TI-83/TI-84: Use the log button to enter log(25)/log(3) and press ENTER. Note: It’s not obvious why entering log(25)/log(3) should be equivalent to 25log3 . We will prove this fact later in this unit. 12. Use the calculator to find the value of each of the logarithms from problem 1. Round

your answers to three decimal places.

a. 25log3 b. 35log2 c. 68log d. )(log 101

3 Answers a. 2.930 b. 5.129 c. 1.833 d. -2.096 13. Solve the following equation: 143743 =−⋅ x . Hint: Start by doing the easy steps, then

switch to “log form.” Use the calculator to write your answer as a decimal.

14. Here’s the solution to the last problem:

143743 =−⋅ x

15043 =⋅ x

504 =x

50log4=x

822.2=x

Use this approach to solve each of the following equations.

a. 95734 =+⋅ x b. 375.150 += x

c. 10072 =x d. 103225 13 =−⋅ +x

e. 5004 22

=−x f. xx 33 10328107 ⋅+=⋅

Answers a. 2.814 b. 6.326 c. 1.183 d. 1.131 e. 546.2± f. 0.282

15. Solve each of the following problems without graphing on your calculator. (You may use the log features of your calculator.)

a. Claire invested $2000 in her bank account earning 2% each year. How many years will it take for her bank account to reach $2100?

b. Arjun’s computer is currently worth $950, but every year it loses 18% of its value.

When will the computer be worth $400?

c. Scott has an investment that earns 7% every year. How many years will it take for the value of his investment to double?

d. A certain element has a half life of 21 days. If the starting amount was 17 kg, how

many days will it take before only 9 kg are left?

Answers a. 2.46 years b. 4.359 years c. 10.245 years d. 19.268 days

16. Complete each of the following tables. x x2 0

1 2

3 4

5 6

7

x x2log

1

2 4

8 16

32 64

128

a. Describe what you observe. Be as specific as possible.

b. Explain what is meant if we say that the function xxf 2)( = has an “add-multiply” property.

c. Does the function xxg 2log)( = also have an “add-multiply” property? Some other property? Explain.

d. Which of the following statements is true: 4343 222 +=⋅ or )4)(3(43 222 =⋅ ?

e. Which of the following statements is true: 16log8log)168(log 222 +=⋅ or 16log8log)168(log 222 ⋅=⋅ ?

f. You should have found that 4343 222 +=⋅ and 16log8log)168(log 222 +=⋅ . Are these statements related? Explain.