Section 7.4: Logarithms
Let's look at the INVERSE of an exponential function.
Case 1: f(x) = ax when a > 1
f(x) = 2x
log22 x
f­1(x) =_________
­3
­2
­1
0
1
2
3
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Now, let's graph both T­Charts on the same graph.
Geometer's Sketchpad File
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Case 2: f(x) = ax when 0< a < 1
f(x) =(1/2)x
log1/2 x
f­1(x) =_________
­3
­2
­1
0
1
2
3
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Now, let's graph both T­Charts on the same graph.
Geometer's Sketchpad File
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2
Now, go back and look at your graphs of your inverses.
A. f(x) = log2 x
1. The input is a value or power?
The output is a value or power?
2. Therefore,
The domain of the function is:
The range of the function is:
3. The graph has an asymptote at:
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4. This function is_________because as x _____, y ____________.
5. End Behavior:
To the right, the graph______________
To the left, the graph_______________
Now, lets go through this process again for case 2.
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B. f(x) = log1/2 x
1. The input is a value or power?
The output is a value or power?
2. Therefore,
The domain of the function is:
The range of the function is:
3. The graph has an asymptote at:
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4. This function is_________because as x _____, y ____________.
5. End Behavior:
To the right, the graph______________
To the left, the graph_______________
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Summary
1. What are the similarities and differences of the 2 types of Logarithmic Functions?
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2. What are the similarities and differences between exponential functions and logarithmic functions?
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So what the heck is a LOGARITHM anyway?
A. A logarithm is an exponent.
Log = Exponent
B. The logarithmic function is the INVERSE of the exponential function.
We just developed this concept in the previous part of this lesson. Now, we will formally define it and work with it.
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C. If f(x) = 2x, (an exponential function), we know f has an inverse because the graph of f(x) = 2x passes the HLT.
Earlier in the lesson, we drew the graph of f­1 by switching the ordered pairs in our 2x table and using y = x, the line of reflection.
We found the inverse GRAPHICALLY.
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D. Now, what about algebraically? To find f­1 of f(x) = 2x:
y = 2x
x = 2y.....What do we do now?
You need to solve for y, but how? **This is why we need logarithms!
So we define y, the exponent, to be the log2 x.
y = log2 x
Therefore,
f(x) = 2x and f­1(x) = log2 x
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http://www.purplemath.com/modules/logs.htm
E. Let's practice conversions from
Exponential Form
Logarithmic Form
A. 23 = 8
log2 8 = 3
log = exponent
B. 34 = 81
log3 81 = 4
C. __________
log2 (1/2) = ­1
D. 32 = 9
____________
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F. Now, practice on your own! Simply rewrite the following:
Exponential Form
Logarithmic Form
1. _________
2. 51 = 5
3. (1/2)­2 = 4
4. _________
5. 45 = 1024
6. _________
7. _________
8. 83 = 512
9. 122 = 144
10. _________
log3 9 = 2
___________
___________
log18 1 = 0
___________
log5 (1/5) = ­1
log36 (1/6) = ­1/2
___________
___________
log14 196 = 2
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G. Evaluating Logs
Note: Your calculator cannot help with the following problems.
LOG on your calculator means a base 10 log (log10)
LN on your calculator means a base e log (loge)
(This is also known as the natural log.)
Keep in mind: Since Logs are exponents, the exponents are your outputs (answers!)
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1. log7 49 = _________
Tips:
7 = 49
2. log2 16 = ________
2 = 16
3. log2 (1/4) = ______
2 = 1/4
4. log7 (1/7) = ______
5. log7 1/7 = ______
6. log
5 1 = _________
7. log12 12 = ________
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8. log2 (1/8) =_________
9. log16 8 = ___________
Hint:
16 = 8
(24) = 23
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Homework: p. 503­505 #3­6, 8­19, 21­29 odd 33, 35, 65­79 odd
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Apr 5­7:46 PM
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Attachments
Section 8­4 Logarithms.gsp
Section 8­4 LogarithmsCase1.gsp
Section 8­4 LogarithmsCase 2.gsp