Unit 9: Logarithms - Wikispaces9+-+Logarithms+Packet.… · GUIDE: properties of logarithmic...

Intermediate Algebra Unit 9: Logarithms


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Objectives: page

introduction to logarithms 2 – 4

logarithms & logarithmic functions 5 – 7

GUIDE: properties of logarithmic functions 8

product & quotient property of logarithms 9 – 11

power property of logarithms 12 – 14

simplifying and solving logarithms 15

solving equations with logarithms 16 – 17

review questions 18 – 19

logarithms & word problems 20 – 23


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Introduction to Logarithms:

Exponential Function )yandxtheswitch(

inverse → Logarithmic Function

y = bx x = by

Rewriting Equations into Exponential Form and Logarithmic Form:


3

Common Logarithm: Change of Base Formula:

Use logarithmic and exponential properties to solve for x:

(1) x8log2 = (2) x41log2 = (3) x256log4 =

(4) 264logx = (5) 3125logx = (6) 316logx =

(7) 0xlog7 = (8) 21xlog81 = (9)

32xlog64 =

(10) 7xlog = (11) x100log = (12) 1101logx −=


4

Practice Examples: (1) FILL IN THE BLANK: A logarithm is an . (2) Given 103 = 1000, rewrite in logarithmic form: log __ 1000 = _____ Write the following equation in exponential form and logarithmic form respectively:

(3) 8134 = (4) 412 2 =−

(5) 3125log5 = (6) 3641log4 −=

Use a calculator and the change of base formula to evaluate each of the following: (7) 8log2 (8) 27log3

(9) 625log5 (10) 91log3


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Logarithms & Logarithmic Functions: Do Now: Write each equation in logarithmic form: Write each equation in exponential form:

(1) 54 = 625

(2) 4917 2 =−

(3) 481log3 =

(4) 216log36 =

Evaluate each expression:

(5) 3log9 (6) 81log2

Solve each equation and check your solution(s):

(7) 23xlog9 = (8) 3xlog

101 −= (9) ( ) 21x2log3 =−

(10) 29logb = (11) ( ) ( )7xlog5x3log 22 +=− (12) ( ) ( )255 x2log1x3log =−


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(13) ( ) ( )2xlog3x2log 66 +=− (14) 21xlog5 = (15) 2121logb =

(16) ( ) ( )1xlog10x4log 22 −=− (17) ( ) xlog6xlog 102

10 =− (18) ( ) 100log36xlog 72

7 =+


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Practice Examples: Write each equation in logarithmic form: Write each equation in exponential form:

(1) 23 = 8

(2) 6418 2 =−

(3) 213log9 =

(4) 2251log5 −=


(5) x25log5 = (6) x1000log10 = (7) x641log4 =

(8) x8log 38 = (9) 5xlog3 = (10) 3xlog

41 =

(11) 213logx = (12) ( ) 212x4log6 =+ (13) ( ) ( )x3log2xlog 33 =+


8

Properties of Logarithmic Functions If b, M, and N are positive real numbers, b ≠ 1, and p and x are real numbers, then:

Definition Examples

01logb = written exponentially: b0 = 1

1blogb = written exponentially: b1 = b

xblog xb = written exponentially: bx = bx

xb xlog b = , where x > 0 710 7log 10 =

NlogMlogMNlog bbb += xlog9logx9log 333 +=

zlogylogyzlog51

51

51

+=

NlogMlogNMlog bbb −=

5log2log52log 444 −=

xlog7logx7log 888 −=

MlogpMlog bp

b = 6logx6log 2

x2 =

ylog4ylog 54

5 =

NlogMlog bb = if and only if M = N )2x5(log)4x3(log 66 +=−

)2x5()4x3( +=−∴

Common Errors:

NlogMlogNlogMlog

bbb

b −≠ NMlogNlogMlog bbb =−

NlogMlog

b

b cannot be simplified

NlogMlog)NM(log bbb +≠+ MNlogNlogMlog bbb =+

)NM(logb + cannot be simplified

Mlogp)M(log bp

b ≠ p

bb MlogMlogp = p

b )M(log cannot be simplified


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Do Now:

Examples:


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Product & Quotient Property of Logarithms:


(1) 10logxlog5log 333 =+ (2) 27log9logxlog 444 =+

(3) 2logx2log16log 101010 =− (4) ( ) 8log5xlog24log 777 =+−


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(5) 7logxlog42log 333 =− (6) 30log5logx3log 222 =+

(7) ( ) ( ) 23xlog21x12log 222 =−−− (8) ( ) ( ) 12xlog2xlog 22 =−−+


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Do Now:

Examples:


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Power Property of Logarithms:


(1) 9logxlog2 55 = (2) xlog24log3 77 =

(3) 49log2116log

41xlog 222 += (4) xlog27log

316log2 101010 =−


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(5) xlog5log24log 999 =+ (6) xlog4log2log3 888 =−

(7) 05log2xlog 22 =+ (8) xlog2log312log10log 5555 +=+


15


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Solving Equations with Logarithms: Do Now: Solve for x:

(1) x125log5 = (2) x64log8 =

Use logarithms to solve for x and, if necessary, round your answer to four decimal places:

(3) 3x = 7 (4) 2x = 5 (5) 14x = 8

(6) 5 x + 1 = 23 (7) 6x = 1.4 (8) 7x – 3 = 5


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(9) 4x = 2 (10) 3x = 80 (11) 9x = 27

(12) 7x = 343 (13) 6x = 127 (14) 12x = 303

(15) 13x = 2839 (16) 2x = 90 (17) 4x = 512

(18) 3x = 5.2 (19) 11x = 153 (20) 10x = 0


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Review Questions: *Study your orange textbook exercises and be sure you can answer these questions: ♦ A logarithm is an _______________________.

♦ Logarithms are used to solve for a variable in the __________________________.

♦ Common (decimal) logs are____________________________________________.

♦ Write the procedure for solving log equations here:

Write each equation in exponential form:

(1) 8log3 2= (2) 2.0log1 5=− Write each equation in log form:

(3) 2749 = (4) 35125 = Solve each equation for x and round your answer to four decimal places: (5) 6.35x = (6) 5.12x =

(7) 214x = (8) 5607 2x =+


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Use logarithmic properties to solve the following equations and check your solution(s):

(9) 32x6log27 = (10) ( ) ( )12xlog3x7log 22 +=−

(11) 216log3164log

21xlog 555 += (12) 4log)x2(log32log 777 =−

(13) 405log5logxlog4 222 =+ (14) 2xlog5log 1010 =+

(15) ( ) ( ) 14log2xlog3xlog 333 =−++ (16) ( ) ( ) 45xlog1xlog 22 =−++


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Logarithms & Word Problems: Do Now: (1) When solving for an unknown exponent, use (2) Solve for x and round your answer to four decimal places: 3.4x = 180.7

(3) A certain car depreciates in value 16% each year. (a) Write an exponential function to model the depreciation of a car that cost

$32,500 when purchased new.

(b) Suppose the car was purchased in 2010. What is the first year the car will be worth less than half its original value? (Solve algebraically.)

(4) Alex invested $2500 at a rate of 2.3% in the bank. If the interest is compounded daily, when will Alex’s money double? Round your answer to the nearest year.


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(5) Solve for x and round your answer to four decimal places: 1.0246x = 4500

(6) (a) Write an equation that shows how much money will be in a savings account that pays 2.75% interest compounded monthly when $500 is invested. (Assuming there are no other withdrawals or deposits.)

(b) Using your equation from part (a), find how long, to the nearest year, the initial

investment of $500 must be left in this account in order for the account to have a value over $795.

(7) Lea bought a car for $38,500. It is expected to depreciate at a rate of 12% per year. After how many years, to the nearest year, will it be worth less than $12,000?

(8) Ms. Pina bought a painting for $5000. It is expected to appreciate in value at a rate of 4% per year. When, to the nearest year, will the painting be worth more than $6300?


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(9) Solve for x. Show your work for full credit. Round your answer to the nearest tenth. 88.95.2 x =

(10) Michael invested $2000 at 4.5% interest compounded quarterly. At this rate, how long, to the nearest year, will take for Michael’s money to triple, assuming there are no other withdrawals or deposits? (Provide an algebraic solution showing all work.)

nt

nr1PA

+=


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(11) A piece of machinery valued at $ 475,000 depreciates at a fixed rate of 7.2% per year. At this rate, after how many years, to the nearest year, will the value of the equipment be below $225,000? (Provide an algebraic solution showing all work.)

(12) You buy an autographed limited edition U2 CD for $20.00 with the understanding that it will increase in value at a steady rate of 2.75% per year. At this rate, how long, to the nearest year, will it take for the CD to reach a value of $100?

(Provide an algebraic solution showing all work.)

CHALLENGE: Solve for x and round your answer to four decimal places:

2 x = 5 x – 2


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