7.5 Logarithm Laws · PDF file 2020-05-08 · PreCalculus 12 2 Logarithm Laws: (for...

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Transcript of 7.5 Logarithm Laws · PDF file 2020-05-08 · PreCalculus 12 2 Logarithm Laws: (for...

  • PreCalculus 12

    1

    7.5 Logarithm Laws Investigate: - Use a calculator to approximate to 4 decimal places:

    1. log 5 + log 6 = log 11 = log 30 =

    log 11 + log 9 = log 20 = log 99 =

    Or, in other words . . . .

    log M + log N =

    Why? We can prove this by:

    Let logcM = x logcN = y

    2. log 35 - log 5 = log 30 = log 7 =

    log 48 - log 4 = log 44 = log 12 =

    Or, in other words . . . .

    log M - log N =

    Once again . . .

    1.48 1.04 1.48 1 996 1.30 1.996

    I log.CM N

    M Ck N CY

    logfM N logfc a logfents se t y

    log MN 109cm t tog n

    0.845 1.478 O 845

    1.079 I 644 1 079

    I log F

    logfF log log Ie 9

    z y loyalty logon loosen

  • PreCalculus 12

    2

    Logarithm Laws: (for any base ! > 0,≠ 1 ()* +, , > 0)

    Example: - Write each expression in terms of individual logarithms of x, y, and z:

    a) b)

    On your Own

    c) d)

    log logc cM N+ =

    log logc cM N- =

    log pc M =

    5log xy z

    3

    log x y z

    6 2

    1 log

    x 3

    7log x

    Product rule log Min

    Quotient Rule log MT

    Power Rule P log M

    n logs 2 logs y logs z 10923 logy log zk

    3 ogre logy I log 2

    A log set

    log I log si flog a 0

    104 2 logs 2e

    O 2109 2e

    21096cal

  • PreCalculus 12

    3

    Example: - Use the laws of logarithms to simplify and evaluate each expression.

    a) b)

    c)

    Example: Write each expression as a single logarithm in simplest form. State the restrictions on the variable.

    a) b)

    6 6 6log 8 log 9 log 2+ - 7log 7 7

    2 2 2 1

    2log 12 log 6 log 27 3

    æ ö- +ç ÷ è ø

    2 7 7 7

    5log log log

    2 x

    x x+ - ( ) ( )25 5log 2 2 log 2 3x x x- - + -

    log 7 t log JI log log 7 t log F'T logo 3G I t E log I It I 127

    o III n n210 12 log6 blog 27 log122 log6 log 27 3

    log.LI2ar3 log o log s

    logy I 131

    109722 1 2 Flogic logs a log a Hogan logan logs ff loyal r logs ET log f n't y

    log z E E

    log at x o

  • PreCalculus 12

    4

    Example: - The pH scale is used to measure the acidity or alkalinity of a solution. The pH of a solution is defined as pH = -log [H+], where [H+] is the hydrogen ion concentration in moles per litre (-./0 ). A neutral solution, such as pure water, has a pH of 7. Solutions with a pH of less than 7 are acidic and solutions with a pH of greater than 7 are basic or alkaline. The closer the pH is to 0, the more acidic the solution is. a) A common ingredient in cola drinks is phosphoric acid, the same ingredient found in many rust removers. A cola drink has a pH of 2.5. Milk has a pH of 6.6. How many times as acidic as milk is a cola drink? b) An apple is 5 times as acidic as a pear. If a pear has a pH of 3.8, then what is the pH of an apple?

    Assignment: P. 441 # 4,5,7-9,12-14 MC1/2

    0

    Cola Milk Ht Cola pit log Ht pit log H H 2 s toyLite 6.6 log Hm

    2 se ogLHe 66 1091hm 63,096K

    I He I 0 Hm as acidic

    take log of both sides Htapple g loglion log s 103 8Ht pear

    10 2 s X tog lo log s.io 3 s

    1 k log s 103 s

    log to 10

    K S 103.8

    Iz 3 I