Chabot Mathematics
§9.4b
Log Base-Change
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Review § 9.4
MTH 55
 Any QUESTIONS About
• §9.4 → Logarithm Properties
 Any QUESTIONS About HomeWork
• §9.4 → HW-46
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Summary of Log Rules
 For any positive numbers M, N,
and a with a ≠ 1
log a ( MN )  log a M  log a N ;
log a M
p
 p log a M ;
M
log a
 log a M  log a N ;
N
k
log a a  k .
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3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Typical Log-Confusion
 Beware that Logs do NOT behave
Algebraically. In General:
log a ( MN )  (log a M )(log a N ),
M log a M
log a

,
N log a N
log a ( M  N )  log a M  log a N ,
log a ( M  N )  log a M  log a N .
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Change 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:
log a x
log x
ln x
log b x 


log a b
log b
ln b
(base a) (base 10) (base e)
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Derive 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
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Example  Evaluate Logs
 Compute log513 by changing to
(a) common logarithms
(b) natural logarithms
 Soln
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log13
a. log 5 13 
log 5
 1.59369
ln13
b. log 5 13 
ln 5
 1.59369
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Example  Evaluate Logs
 Use the change-of-base formula to
calculate log512.
• Round the answer to four decimal places
 Solution
 Check
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log12
log 5 12 
log5
 1.5440
51.5440  12.0009  12

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Example  Evaluate Logs
 Find log37 using the change-of-base
formula
log10 7
 Solution log3 7 
log10 3
0.84509804

0.47712125
Substituting into
log a M
logb M 
.
log a b
1.7712
3
 7.000
 1.7712
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Example  Swamp Fever
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10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Example  Swamp Fever
This does NOT = Log3
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Logs with Exponential Bases
log b a
log b k a 
k
log b b
log b a

k log b b
1
log b k a  log b a
k
 Consider an example where k = −1
1
log1 b a  logb1 a  logb a   logb a
1
 For a, b >0,
and k ≠ 0
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Example  Evaluate Logs
 Find the value of each expression
withOUT using a calculator
3
a. log 5 5
b. log1 3 3
c. 7
log1 7 5
1
3
 Solution a. log 5 3 5  log 5 5
1
 log 5 5
3
1

3
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13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Example  Evaluate Logs
 Solution:
b. log1 3 3
b. log1 3 3  log 31 3
  log 3 3
 1
c. 7
c. 7
log1 7 5
log1 7 5
7
log
7
 log 7 5

 7
7 1
5
1

5
14

log 7 5 1
1
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5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Example  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   ae
b0 
 ae  a 1  a
0
 So a = 2 and f(x) = 2ebx . Now
substitute (3, 8) in to the equation.
8  f 3  2e
b3
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 2e
3b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Example  Curve Fit
 Now find b by
Taking the
Natural Log
of Both Sides
of the Eqn
8  2e
4e
ln 4  3b
1
b  ln 4
3
3b
 Thus the
function
that will fit the Curve
aebx
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3b
f x   2e
1

 ln 4  x
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
WhiteBoard Work
 Problems From §9.4 Exercise Set
• 70, 74, 76, 78, 80, 82
 Log Tables
from John
Napier, Mirifici
logarithmorum
canonis descriptio,
Edinburgh, 1614.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
All Done for Today
Logarithm
Properties
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt