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Transcript of [email protected] MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 1 Bruce Mayer, PE Chabot College...
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§9.4b§9.4bLog Base-ChangeLog Base-Change
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §9.4 → Logarithm Properties
Any QUESTIONS About HomeWork• §9.4 → HW-46
9.4 MTH 55
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt3
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
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt4
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
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt5
Bruce Mayer, PE Chabot College Mathematics
Change of Base RuleChange of Base Rule
Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then logbx can be converted to a different base as follows:
logb x loga x
loga b
log x
logb
ln x
lnb
(base a) (base 10) (base e)
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt6
Bruce Mayer, PE Chabot College Mathematics
Derive Change of Base RuleDerive Change of Base Rule
Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt7
Bruce Mayer, PE Chabot College Mathematics
Example Example Evaluate Logs Evaluate Logs
Compute log513 by changing to (a) common logarithms (b) natural logarithms
Soln
b. log5 13 ln13
ln 51.59369
a. log5 13 log13
log 5
1.59369
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt8
Bruce Mayer, PE Chabot College Mathematics
Use the change-of-base formula to calculate log512.
• Round the answer to four decimal places
Solution
Example Example Evaluate Logs Evaluate Logs
5
log12log 12
log5
1.5440
1.54405 12.0009 12 Check
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt9
Bruce Mayer, PE Chabot College Mathematics
Find log37 using the change-of-base formula
Solution
Example Example Evaluate Logs Evaluate Logs
Substituting into log
log .loga
ba
MM
b
0.84509804
0.47712125
1.7712
103
10
log 7log 7
log 3
000.73 7712.1
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt10
Bruce Mayer, PE Chabot College Mathematics
Example Example Swamp Fever Swamp Fever
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Bruce Mayer, PE Chabot College Mathematics
Example Example Swamp Fever Swamp Fever
This does NOT = Log3
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt12
Bruce Mayer, PE Chabot College Mathematics
Logs with Exponential BasesLogs with Exponential Bases
For a, b >0, and k ≠ 0
logbka
logb a
logb bk
logb a
k logb b
logbka
1
klogb a
Consider an example where k = −1
log1 b a logb 1 a
1
1logb a logb a
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt13
Bruce Mayer, PE Chabot College Mathematics
Example Example Evaluate Logs Evaluate Logs
Find the value of each expression withOUT using a calculator
a. log5 53 b. log1 3 3 c. 7log1 7 5
Solution a. log5 53 log5 51
3
1
3log5 5
1
3
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt14
Bruce Mayer, PE Chabot College Mathematics
Example Example Evaluate Logs Evaluate Logs
Solution: b. log1 3 3 c. 7log1 7 5
b. log1 3 3 log3 1 3
log3 3
1
c. 7log1 7 5 7
log7 1 5
7 log7 5
7log7 5 1
5 1
1
5
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt15
Bruce Mayer, PE Chabot College Mathematics
Example Example Curve Fit Curve Fit
Find the exponential function of the form f(x) = aebx that passes through the points (0, 2) and (3, 8)
Solution: Substitute (0, 2) into f(x) = aebx
2 f 0 aeb 0 ae0 a1 a
So a = 2 and f(x) = 2ebx . Now substitute (3, 8) in to the equation.
8 f 3 2eb 3 2e3b
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt16
Bruce Mayer, PE Chabot College Mathematics
Example Example Curve Fit Curve Fit
Now find b by Taking the Natural Logof Both Sidesof the Eqn
8 2e3b
4 e3b
ln 4 3b
b 1
3ln 4
f x 2e1
3ln 4
x Thus the aebx function
that will fit the Curve
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt17
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §9.4 Exercise Set• 70, 74, 76, 78, 80, 82
Log Tablesfrom John Napier, Mirifici logarithmorum canonis descriptio,Edinburgh, 1614.
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt18
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
LogarithmProperties
[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt19
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22