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 1.481 996 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

z yloyalty 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 =

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

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 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

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 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