[email protected] mth55_lec-62_sec_9-4a_log_rules.ppt 1 bruce mayer, pe chabot college...
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[email protected] • MTH55_Lec-62_sec_9-4a_Log_Rules.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§9.4a§9.4aLogarithm RulesLogarithm Rules
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Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §9.3 → Common & Natural Logs
Any QUESTIONS About HomeWork• §9.3 → HW-45
9.3 MTH 55
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Bruce Mayer, PE Chabot College Mathematics
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.
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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)
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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”
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Bruce Mayer, PE Chabot College Mathematics
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
[email protected] • MTH55_Lec-62_sec_9-4a_Log_Rules.ppt11
Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
Example Example Use The Rules Use The Rules
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
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Bruce Mayer, PE Chabot College Mathematics
Example Example Use The Rules Use The Rules
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
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Bruce Mayer, PE Chabot College Mathematics
Example Example Use The Rules Use The Rules
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
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Bruce Mayer, PE Chabot College Mathematics
Example Example Use The Rules Use The Rules
Solnb)
71
log3 b
xy
z
71log log
3 b bxy z
1log log 7log
3 b b bx y z
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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)
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Bruce Mayer, PE Chabot College Mathematics
Example Example Expand by Log Rules Expand by Log Rules
Solution b)
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
Example Example Condense Logs Condense Logs
Solution a)
Solution b)
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Bruce Mayer, PE Chabot College Mathematics
Example Example Condense Logs Condense Logs
Solution c)
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Bruce Mayer, PE Chabot College Mathematics
Example Example Condense Logs Condense Logs
Solution d)
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Bruce Mayer, PE Chabot College Mathematics
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
[email protected] • MTH55_Lec-62_sec_9-4a_Log_Rules.ppt24
Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
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
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §9.4 Exercise Set•
24, 30, 36, 58, 60
CondenseLogarithm
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Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
MathematicalAssociationLog Poster
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
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