[email protected] MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.

[email protected] • MTH55_Lec-62_sec_9-4a_Log_Rules.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§9.4a§9.4aLogarithm RulesLogarithm Rules



Review §Review §

Any QUESTIONS About• §9.3 → Common & Natural Logs

Any QUESTIONS About HomeWork• §9.3 → HW-45

9.3 MTH 55



Product Rule for LogarithmsProduct Rule for Logarithms

Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. Then the PRODUCT Rule

loga MN loga M loga N

That is, The logarithm of the product of two (or more) numbers is the sum of the logarithms of the numbers.



Quotient Rule for LogarithmsQuotient Rule for Logarithms

Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. Then the QUOTIENT Rule

That is, The logarithm of the quotient of two (or more) numbers is the difference of the logarithms of the numbers

loga

M

N

loga M loga N



Power Rule for LogarithmsPower Rule for Logarithms

Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. Then the POWER Rule

That is, The logarithm of a number to the power r is r times the logarithm of the number.

loga M r r loga M



Example Example Product Rule Product Rule

Express as an equivalent expression that is a single logarithm: log3(9∙27)

Solutionlog3(9·27) = log39 + log327.

• As a Check note that

log3(9·27) = log3243 = 5 35 = 243

• And that

log39 + log327 = 2 + 3 = 5. 32 = 9 and 33 = 27



Example Example Product Rule Product Rule

Express as an equivalent expression that is a single logarithm: loga6 + loga7

Solution

= loga(42). Using the product rule for logarithms

loga6 + loga7 = loga(6·7)



Example Example Quotient Rule Quotient Rule

Express as an equivalent expression that is a single logarithm: log3(9/y)

Solution

log3(9/y) = log39 – log3y. Using the quotient rule for logarithms



Example Example Quotient Rule Quotient Rule

Express as an equivalent expression that is a single logarithm: loga6 − loga7

Solution

loga6 – loga7 = loga(6/7) Using the

quotient rule for logarithms “in reverse”



Example Example Power Rule Power Rule

Use the power rule to write an equivalent expression that is a product:

a) loga6−3 4b) log .x

Solution

= log4x1/2

Using the power rule for logarithms a) loga6−3 = −3loga6

4b) log x

= ½ log4x Using the power

rule for logarithms



Example Example Use The Rules Use The Rules

Given that log5z = 3 and log5y = 2, evaluate each expression.

a. log5 yz b. log5 125y7

c. log5

z

yd. log5 z

1

30 y5

a. log5 yz log5 y log5 z

2 3

5

Solution




Solution

Soln

c. log5

z

ylog5

z

y

1

2

1

2log5 z log5 y

1

23 2 1

2

b. log5 125y7 log5 125 log5 y7

log5 53 7 log5 y

3 7 2 17




Soln

d. log5 z1

30 y5

log5 z

1

30 log5 y5

1

30log5 z 5 log5 y

1

303 5 2

0.110

10.1




Express as an equivalent expression using individual logarithms of x, y, & z

Solna)

334 7

a) log b) logbx xy

yz z

= log4x3 – log4 yz

= 3log4x – log4 yz

= 3log4x – (log4 y + log4z)

= 3log4x –log4 y – log4z




Solnb)

71

log3 b

xy

z

71log log

3 b bxy z

1log log 7log

3 b b bx y z



CaveatCaveat on Log Rules on Log Rules

Because the product and quotient rules replace one term with two, it is often best to use the rules within parentheses, as in the previous example

1log log 7log

3 b b bx y z

71

log3 b

xy

z



Example Example Expand by Log Rules Expand by Log Rules

Write the expressions in expanded form

a. log2

x2 x 1 3

2x 1 4 b. logc x3y2z5

Solution a)



Example Example Expand by Log Rules Expand by Log Rules

Solution b)



Example Example Condense Logs Condense Logs

Write the expressions in condensed forma. log 3x log 4y

b. 2 ln x 1

2ln x2 1

c. 2 log2 5 log2 9 log2 75

d. 1

3ln x ln x 1 ln x2 1




Solution a)

Solution b)




Solution c)




Solution d)



Log of Base to ExponentLog of Base to Exponent

For any Base a

That is, the logarithm, base a, of a to an exponent is the exponent



Example Example Log Base-to-Exp Log Base-to-Exp

Simplify: a) log668 b) log33−3.4

Solution a)

log668 =8 8 is the exponent to which you raise 6 in order to get 68.

Solution b)

log33−3.4 = −3.4



Summary of Log RulesSummary of Log Rules

For any positive numbers M, N, and a with a ≠ 1

log log log ;a a aM

M NN

log log ;pa aM p M

log .ka a k

log ( ) log log ;a a aMN M N



Typical Log-ConfusionTypical Log-Confusion

BewareBeware that Logs do NOT behave Algebraically. In General:

loglog ,

loga

aa

MM

N N

log ( ) (log )(log ),a a aMN M N

log ( ) log log ,a a aM N M N

log ( ) log log .a a aM N M N



WhiteBoard WorkWhiteBoard Work

Problems From §9.4 Exercise Set•

24, 30, 36, 58, 60

CondenseLogarithm



All Done for TodayAll Done for Today

MathematicalAssociationLog Poster



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

[email protected] MTH55_Lec-62_sec_9-4a_Log_Rules.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.

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