8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
8­3 Day 2
Logarithmic Functions as Inverses
IMPALAPALOOZA
Objective: Graph logarithmic functions.
Mar 22­10:20 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Check Skills You'll Need
Graph.
1) y = 5x
2) y = 2x + 4
3) y = 3x + 2 ­ 1
Mar 22­10:27 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Finding Inverse Functions
How do we find the inverse of y = 3x?
Step 1: Write the exponential equation as a logarithm.
y = 3x is equal to log3 y = x
Step 2: Interchange x and y. (because we switch the domain and range to get the inverse)
log3 y = x
log3 x = y
Therefore, the inverse of y = 3x is y = log3 x.
Mar 31­6:52 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Try one!
Find the inverse of y = (1/2)x.
Mar 31­7:07 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Logarithms and Exponential Functions are inverses. Therefore, we know that the graph of a logarithm is the graph of an exponential function reflected over the line y = x.
Mar 22­10:22 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Graph y = log2 x.
Step 1: Since y = 2x is the inverse of y = log2 x, start by graphing it.
Mar 22­10:23 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Graph y = log2 x.
Step 1: Since y = 2x is the inverse of y = log2 x, start by graphing it.
Step 2: Draw y = x.
Mar 22­10:23 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Graph y = log2 x.
Step 1: Since y = 2x is the inverse of y = log2 x, start by graphing it.
Step 2: Draw y = x.
Step 3: Choose a few points on y = 2x.
Reverse the coordinates and plot the points of y = log2 x. Mar 22­10:23 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
We have graphed exponential functions using a t­table such as the ones below.
y = bx
It's inverse:
y = logb x
x y
x y
1/b ­1
1 0
b 1
­1 1/b
0 1
1 b
The points on the graph of y = logb x are:
(1/b, ­1), (1, 0) and (b, 1).
Mar 22­10:23 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Logarithmic Properties
f (x) = logb x
0 < b < 1
(b,1)
1) The Domain : { x | x > 0}
Range : { y | y = (All Reals)}
(1,0)
2) There are no y intercepts and the x intercept is 1.
(1/b,­1)
3) The y axis (x=0) is a vertical asymptote as x 0.
4) f(x) = logb x , 0 < b < 1, is an decreasing function and is one­to­one.
5) The graph of f(x) contains the points (b,1), (1,0), (1/b,­1).
Jan 11­4:01 PM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Logarithmic Properties
f (x) = logb x
b > 1
(b,1)
(1,0)
1) The Domain : { x | x > 0}
Range : { y | y = (All Reals)}
(1/b,­1)
2) There are no y intercepts and the x intercept is 1.
3) The y axis (x=0) is a vertical asymptote as x 0.
4) f(x) = logb x , b > 1, is an increasing function and is one­to­one.
5) The graph of f(x) contains the points (b,1), (1,0), (1/b,­1).
Jan 11­4:01 PM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Graph y = log1/2 x.
The points on the graph of y = logb x are:
(1/b, ­1), (1, 0) and (b, 1).
Domain:
Range:
Asymptote:
Mar 22­10:23 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Graph y = log3 x.
Domain:
Range:
Asymptote:
Mar 22­10:23 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Mar 22­10:24 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Domain:
Range:
Asymptote:
Graph y = log6 (x ­ 2) + 3.
Step 1: Graph log6 x.
Step 2: Shift the graph right 2 and up 3.
Mar 22­10:24 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Graph y = log1/2 (x + 3).
Domain:
Range:
Asymptote:
Mar 22­10:25 AM
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8­3 Day 2 Logarithmic Functions as Inverses
April 01, 2009
Homework
8­1 through 8­3 Review
Worksheet
Mar 22­10:25 AM
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