[email protected] MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§9.4b§9.4bLog Base-ChangeLog Base-Change



Review §Review §

Any QUESTIONS About• §9.4 → Logarithm Properties

Any QUESTIONS About HomeWork• §9.4 → HW-46

9.4 MTH 55



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



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)



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



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



Use the change-of-base formula to calculate log512.

• Round the answer to four decimal places

Solution


5

log12log 12

log5

1.5440

1.54405 12.0009 12 Check



Find log37 using the change-of-base formula

Solution


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



Example Example Swamp Fever Swamp Fever



Example Example Swamp Fever Swamp Fever

This does NOT = Log3



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




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




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



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



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



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.



All Done for TodayAll Done for Today

LogarithmProperties



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

Documents

[email protected] MTH55_Lec-63_sec_9-4b_Log_Change_Base.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical