sfm productions presents: another saga in your continuing pre-calculus experience! 3.2logarithmic...
TRANSCRIPT
SFM Productions Presents:
Another saga in your continuingPre-Calculus experience!
3.2 Logarithmic Functions and their Graphs
p234 #7-31, 37-41, 51-65, 85-91, 95, 97
Homework for section 3.2
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
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8
9
10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
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9
10
0
exponential
horizontal
Asymptote
y = 0
logarithmic
vertical
asymptote
x = 0
A logarithmic function with base “a”:
is denoted by:
whe( ) re x 0
and 0
log
1af x x
a
logay x if and only if: yx a
logay x yx a
loga x y yx a
A logarithm is an exponent.
A an exponent.
logarithm
is
A an exponent.
logarithm
is
A an exponent.
logarithm
is
A an exponent.
logarithm
is
A an exponent.
logarithm
is
loga x y yx a
logarithm
is exponent.
3log 27 3
loga x y yx aThe two equationsare equivalent…
Use one to solve the other…and use the other to solve the one…depending upon which one you
need to solve.
2log 32 5 is the same as: 52 322log 32 5 52 32
is the same as: 33 27
is the same as:4log 2 12 12
4 2
is the same as: 10
1log
1002 2 1
10100
is the same as:3log 1 0 03 1
is the same as:2log 2 1 12 2
3log 27 3 33 27
4log 2 12 12
4 2
10
1log
1002 2 1
10100
3log 1 0 03 1
2log 2 1 12 2
Properties of Common Logarithms
10
Common log is also known as
l
"lo
og
g base 0"
log
1
x x
log 1 0a 0because: 1a
log 1a a 1because: a a
log xa a x
loga xa x
because: x xa alogarithmic
exponential
because: log loga ax x
All this stuff works with e and ln, too.
Properties of Natural Logarithms
Natural log is also known as "log base
ln l g
e"
o ex x
ln 1 0 0because: 1a
ln 1e 1because: e e
ln xe x
ln xe x
because: x xe elogarithmic
exponentialbecause: ln lnx x
Another Property of Common and Natural Logarithms
I f log loga ax y then: x y
I f x y lothen g lo g: a ax y
I f ln lnx y then: x y
I f x y then: ln n lx y
I f x y then: yxa a
I f x y then: e yx e
For all: f(x) = logax
Increasing:
Decreasing
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
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-4
-3
-2
-1
1
2
3
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5
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8
9
10
0
Domain:Range:
VA:X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
Intercept:
0,
,
1, 0
0x
Shiftingf(x) = log
2x
f(x) = log2x + 3
f(x) = log2x - 4
What is new asymptote???
What is new asymptote???
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0
Shiftingf(x) = log
2x
f(x) = log2(x + 3)
f(x) = log2 (x - 4)
What is new asymptote???
What is new asymptote???
X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
0X
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
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5
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8
9
10
0
Domain
Your favorite…or is it mine???
On your calculators, do:
10log 10
102 log 2 .5
10log 2
10log 0
What can you deduce from this???
You can’t take the log of a negative number, or 0.
ln 2
ln 0
Common or Natural
NCD
Finding domains of log functions…
10log 2f x x defined only if: 2 0x
domai : 2,n
ln 2f x x defined only if: 2 0x
domain: , 2
2lnf x x 2defined only if: 0x
, 0 domai ANDn 0,:
Go! Do!