4.4 Logarithmic Functions 2011 April 25, 2011
Warm­up:
Sketch the graph of each function.
1. y = x
2. y = x2
3. y = x3
4. y = |x|
5. y = √x
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4.4 Logarithmic Functions 2011 April 25, 2011
4.4 Logarithmic Functions
This is an old book of common logarithms. Yes you love your calculator.
Objective:
Write exponential functions as logarithms AND write logarithms as exponential functions.
Graph and find the domain of logarithmic functions.
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4.4 Logarithmic Functions 2011 April 25, 2011
Logarithmic Functions
A logarithm of the base a, where a > 0 and a ≠ 1 can be written:
y = loga x
"y equals log base a of x"
Definition:
y = logax if and only if x = ay
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4.4 Logarithmic Functions 2011 April 25, 2011
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4.4 Logarithmic Functions 2011 April 25, 2011
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4.4 Logarithmic Functions 2011 April 25, 2011
Changing Logarithms to Exponential Functions
y = loga x 2 = log8 k x = ay
k = 82
3 = logd 27 27 = d3 x = log2 16 16 = 2x 6
4.4 Logarithmic Functions 2011 April 25, 2011
Changing Exponential Functions to Logarithms
x = ay y = loga x 34 = 3x x = log3 34 117.39 = 1.1x x = log1.1 117.39
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4.4 Logarithmic Functions 2011 April 25, 2011
Solving Logarithms y = loga x x = ay
3 = log6 x 63 = x x = 216 2 = logx(1/81) x2 = 1/81 x =1/9
log7 2401 log7 2401 = x
2401 = 7x 2401 = 74
x = 4
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4.4 Logarithmic Functions 2011 30 = log6 6x log2 (1/32) April 25, 2011
30 = x log6 6 30 = x (1) 30 = x log2 (1/32) = x 2x = 1/32 2­5 = 1/32
x = ­5
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4.4 Logarithmic Functions 2011 April 25, 2011
Logarithmic Functions
y = loga x if and only if x = a
y
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4.4 Logarithmic Functions 2011 Review:
April 25, 2011
y = 3x
x
1. Graph y = 3 .
2. Find the inverse of y = 3x.
x y
3. Graph the inverse of y = 3x.
x y
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4.4 Logarithmic Functions 2011 April 25, 2011
Logarithmic Properties
f (x) = loga x 0 < a < 1
1) Domain and Range
D: x > 0
R: y = all real #s
2) Intercepts
x intercept is 1 (1, 0)
no y intercepts 3) Vertical Asymptote
(as x ⇒ 0)
VA: x = 0
4) Categorizing the Function
Decreasing Function
One­to­one
5) Points on the Graph
(a, 1) (1, 0) (1/a, ­1)
(a,1)
(1,0)
(1/a,­1)
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4.4 Logarithmic Functions 2011 April 25, 2011
Logarithmic Properties
f (x) = loga x
1) Domain and Range
D: x > 0
R: y = all real #s
2) Intercepts
x intercept is 1 (1, 0)
no y intercepts 3) Vertical Asymptote
(as x ⇒ 0)
VA: x = 0
4) Categorizing the Function
Increasing Function
One­to­one
5) Points on the Graph
(a, 1) (1, 0) (1/a, ­1)
a > 1
(a,1)
(1,0)
(1/a,­1)
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4.4 Logarithmic Functions 2011 April 25, 2011
Where did logarithm come from?
It was Exponential Functions fault!
Logarithm is the inverse of Exponential.
y = loga (x) and y = ax for a > 1
y = loga (x) and y = ax for 0 < a < 1
Remember, logarithms and exponents are inverses.
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4.4 Logarithmic Functions 2011 April 25, 2011
Use transformations to graph the function.
Determine the domain, range and vertical asymptote of the function.
1.
f(x) = 3 + log(x ­ 4)
Hint: a > 1 so the graph includes (a, 1), (1, 0) & (1/a, ­1)
D:
R:
VA:
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4.4 Logarithmic Functions 2011 April 25, 2011
Use transformations to graph the function.
Determine the domain, range and vertical asymptote of the function.
2.
f(x) = 4log(x + 2)
Hint: a > 1 so the graph includes (a, 1), (1, 0) & (1/a, ­1)
D:
R:
VA:
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4.4 Logarithmic Functions 2011 April 25, 2011
Use transformations to graph the function.
Determine the domain, range and vertical asymptote of the function.
3.
f(x) = 3 + ln(x)
Hint: a > 1 so the graph includes (a, 1), (1, 0) & (1/a, ­1)
D:
R:
VA:
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4.4 Logarithmic Functions 2011 April 25, 2011
Homework:
page 296 (10 ­ 32 even, 34 ­ 45, 57 ­ 60, 67 ­ 74, 76, 77, 84, 85, 91 ­ 102)
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4.4 Logarithmic Functions 2011 April 25, 2011
When are logarithms used today?
The decibel is the unit of volume from a speaker!
We all love our i­pods and music the decibel system is based on Logarithms!!
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