Ch 3 Logs bl 3.notebook
April 22, 2013
Properties of Logs
Product Rule
logb(RS) = logb(R) + logb(S)
Quotient Rule logb(R/S) = logb(R) - logb(S)
Power Rule
logb(Rc) = c logb(R)
Apr 14­10:11 AM
Proof
logb(RS) =logb(R) + logb(S)
RS = bxby
= b(x+y)
logb(RS) = x + y
= logb(R) + logb(S)
Apr 14­11:10 AM
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Ch 3 Logs bl 3.notebook
April 22, 2013
Proof
logb(R/S) =logb(R) - logb(S)
R/S = bx/by
= b(x-y)
logb(R/S) = x - y
= logb(R) - logb(S)
Apr 14­11:14 AM
Proof
logb(Rc) =c logb(R)
Let y = logbRc
= logb(R*R.........*R)}(c times)
= c logb(R)
Change of base
let y = logbR
by = R
log by =log R
y log by = log R
y = log R / log by = logbR
log3 16 = log 16 / log 3 or
log3 16 = ln 16 / ln 3
Apr 14­11:16 AM
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Ch 3 Logs bl 3.notebook
April 22, 2013
Properties of Natural Logs -LN
Product Rule
ln(RS) = ln(R) + ln(S)
Quotient Rule ln(R/S) = ln(R) - ln(S)
Power Rule
ln(Rc) = c ln(R)
Apr 14­11:35 AM
To solve an equation that has a variable in
the exponent
Isolate the variable term
Then take the log of both sides
(If the base is e, take ln of both sides)
Use the power rule to write x log #
Solve for x
5x = 7
log 5x = log 7
x log 5 = log 7
x = log 7 / log 5
Apr 16­10:03 AM
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Ch 3 Logs bl 3.notebook
April 22, 2013
Solve the equation
4x = 7
ln 4x = ln 7 (take ln of both sides)
xln 4 = ln 7 (use power rule)
x
= ln 7 / ln 4 (divide by ln 4)
Apr 14­11:56 AM
Example 1
32x+1 = 15
log 32x+1 = log 15
(2x+1) log 3 = log 15
2x +1 = log 15 / log 3
2x = (
) -1
x = (
) /2
Example 2
8 + 2ex = 12
2ex = 4
ex = 2
ln ex = ln 2
x
= ln 2
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Ch 3 Logs bl 3.notebook
April 22, 2013
Apr 14­1:49 PM
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Ch 3 Logs bl 3.notebook
April 22, 2013
Rewriting a log of a product
log (8xy4) = log8 + log x + log y4
log 23 + log x + log y4
3 log 2 + log x + 4 log y
Condensing a Logarithmic Function
5 ln x - 2 ln (xy) = ln x5-ln (x2y2)
= ln ((x5)/(x2y2))
= ln (x3/y2)
Apr 14­11:46 AM
Do p 299 1-14
15-18
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Ch 3 Logs bl 3.notebook
April 22, 2013
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Ch 3 Logs bl 3.notebook
April 22, 2013
Homework
p 300
# 23,26,29,31,34,35,
39,40,43,44
Apr 14­2:15 PM
How to graph a log function
Remember that since a0 = 1
so the log 1 = 0
ax has a horizontal asymptote of
y=0
so the graph of a log must have a
vertical asymptote at x = 0
The rules of shifts and stretches
still exist.
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Ch 3 Logs bl 3.notebook
April 22, 2013
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Ch 3 Logs bl 3.notebook
April 22, 2013
Apr 14­2:12 PM
Homework / classwork
FMC p 134 #6,8,11,12,18,20,30
p 138 # 5-8, 13-20
Homework
p 313 # 1-15 odd
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Ch 3 Logs bl 3.notebook
April 22, 2013
Evaluate without a calc
6. a) log(log 10)
b) √log 100 -log√100
c. log(√10∛105√10)
d. 1000 log 3
e. .01log 2
f. 1/ (log(1/log(10√10))
Apr 14­2:31 PM
#8 x= log A, y = log B
Write an expression in terms of x and y
a. log(AB)
b. log(A3*√B)
c log (A - B)
d. log(A) / Log (B)
e. Log (A/B)
f. AB
Apr 14­2:39 PM
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Ch 3 Logs bl 3.notebook
April 22, 2013
Let p = log m and q= log n Write
the following expressin in terms
of p and q w/o logs
a. m
b. n3
c. log mn3
d. log √m
Apr 14­2:45 PM
12. 5(1.031)x = 8
18. log (1-x) - log (1+x) =2
20. bx = c
30. 58e(4t+1)= 30
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Ch 3 Logs bl 3.notebook
graph
y = 2 3x + 1
y = -e-x
April 22, 2013
y = log(x-4)
y = ln(x + 1)
Apr 14­2:57 PM
Find the domain
13. ln(x2)
14. (ln x)2
15. ln(ln x)
16. ln(x-3)
Apr 14­2:59 PM
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Ch 3 Logs bl 3.notebook
April 22, 2013
Continuous Compounding
Finance
A = Pert
A = P (1 + r/n)nt
A= Annual Yield
r = interest rate
n = number of times it
is compounded per year
t = number of years
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Ch 3 Logs bl 3.notebook
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Ch 3 Logs bl 3.notebook
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