7.5 Logarithm Laws · 2020-05-08 · PreCalculus 12 2 Logarithm Laws: (for any base !>0,≠ 1 ()*...
Transcript of 7.5 Logarithm Laws · 2020-05-08 · PreCalculus 12 2 Logarithm Laws: (for any base !>0,≠ 1 ()*...
PreCalculus 12
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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 1.481 996 1.30 1.996
I log.CM N
M Ck N CY
logfM N logfc a
logfentsse 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
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PreCalculus 12
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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 =
5logxyz
3
logxy z
6 2
1log
x3
7log x
Product rule log Min
Quotient Rule log MT
Power Rule P log M
nlogs 2 logs y logs z 10923 logy log zk
3 ogre logy I log 2
Alog set
log I log si flog a
0104 2 logs 2e
O 2109 2e
21096cal
PreCalculus 12
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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 21
2log 12 log 6 log 273
æ ö- +ç ÷è ø
2 77 7
5loglog log
2x
x x+ - ( ) ( )25 5log 2 2 log 2 3x x x- - + -
log 7 t log JI
log log 7 t log F'Tlogo 3G I t Elog I It I127
o IIIn n
210 12 log6 blog 27log122 log6 log 27 3
log.LI2ar3log o
log s
logy I131
109722 1 2 Flogic logs a
log a Hogan logan logs ffloyal r logs ETlog f n't y
log z E E
log at x o
PreCalculus 12
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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 Colapit log Ht pit log H H2 s toyLite 6.6 log Hm
2 se ogLHe 66 1091hm 63,096K
I He I 0 Hm as acidic
take log of both sidesHtapple g loglion log s 103 8Ht pear
102 s X tog lo log s.io 3 s
1 k log s 103 s
log to10
K S 103.8
Iz 3 I