l ogarithmic f unctions section 8.4. 8.4 l ogarithmic f unctions objectives: 1.write logarithmic...
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8.4 LOGARITHMIC FUNCTIONS
Objectives:
1. Write logarithmic functions in exponential form and back.
2. Evaluate logs with and without a calculator.
3. Evaluate logarithmic functions.
4. Understand logs and inverses.
5. Graph logarithmic functions.
Vocabulary:
logarithm, common logarithm, natural logarithm
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In Section 8.3, we learned that if the interest of a bank account is 5% compounded, then the
total asset after t years is described by:
Yearly: At = P (1 + 0.05 )t
Monthly: At = P (1 + 0.05 / 12)12·t
Daily: At = P (1 + 0.05 / 365)365·t
Continuously: At = P e 0.05·t
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In each case, as long as we know the time, t, we can calculate the final (total) asset:
Yearly: A5 = P (1 + 0.05 )5
Monthly: A10 = P (1 + 0.05 / 12)12·10
Daily: A2 = P (1 + 0.05 / 365)365·2
Continuously: A6 = P e 0.05·6
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Now we would like to ask a reverse question:
How long does the initial deposit (investment) take to reach the target asset value?
Yearly: 2000 = 1200 (1 + 0.05 )t
LET’S THINK
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LOGS TO THE RESCUE
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O S W E G OINTRODUCING…
O S W E G Ohich
xponent
oes
n
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EVALUATE THE EXPRESSIONS
Think: “Which exponent goes on 2 to give me 8?”3
2
3
0
Sorry, but “wego” does not really exist! In math, we use “logarithms.” The problems above would be written with the word “log” instead of “wego.”
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EVALUATE THE EXPRESSIONS
4
2
-2
-3
6Which Exponent
Goes On…
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SPECIAL LOGARITHM VALUES
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Definition: Logarithm of y with base b
Let b and y be positive numbers, and b ≠ 1.
Then, logby = x if and only if y = bx.
Definition: Exponential Function
The function is of the form: f(x) = a · bx, where a ≠ 0, b > 0 and b ≠ 1.
REMEMBER THIS…?
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REWRITING LOGARITHMIC EQUATIONS
Logarithmic Form Exponential Form
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COMMON NOTATION
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EVALUATING COMMON & NATURAL LOGS
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Examples: Evaluate the common and natural logarithms.
a) log4
b) ln(1/5)
c) lne-3
d) log(1/1000)
0.602
-1.609
1
-3
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Practice: Evaluate the common and natural logarithm.
a) ln0.25
b) log3.8
c) ln3
d) lne2007
0.845
-1.386
0.580
1
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pg. 490 #16-19, 24-30, 36, 37
HOMEWORK
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LOGARITHMIC FUNCTIONSSection 8.4 (Day 2)
RULE!
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8.4 LOGARITHMIC FUNCTIONS
Objectives:
1. Write logarithmic functions in exponential form and back.
2. Evaluate logs with and without a calculator.
3. Evaluate logarithmic functions.
4. Understand logs and inverses.
5. Graph logarithmic functions.
Vocabulary:
logarithm, common logarithm, natural logarithm
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From the definition of a logarithm, we noticed that the logarithmic function, g(x) = logbx, is the inverse of the exponential function f(x) = bx.
Recall:How do we verify if two functions are inverses?
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WHAT DOES THIS MEAN?
This means that they offset each other, or they “undo” each other.
These two functions are inverses to each other.
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USING INVERSES: SIMPLIFY THE EXPRESSION
x
x
x
x
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USING INVERSES: SIMPLIFY THE EXPRESSION
x
2x
2x
3x
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HOW DO WE FIND INVERSES?
1. Switch x and y.
2. Solve for y.
3. KAPOOYA! DONE!
4. Check using composition because we are diligent students.
In General…
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LET’S LOOK AT THE SPECIFICS…
In General…
1. Switch x and y.
2. Solve for y.
3. KAPOOYA! DONE!
4. Check using composition because we are diligent students.
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LET’S LOOK AT THE SPECIFICS…
In General…
1. Switch x and y.
2. Solve for y.
3. KAPOOYA! DONE!
4. Check using composition because we are diligent students.
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Examples: Find the inverse of the function
a) y = log8x
b) y = ln(x – 3)
Answers:a) y = 8x
b) y = ex + 3
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Practice: Find the inverse of
a) y = log2/5x
b) y = ln(2x – 10)
Answers:a) y = (2/5)x
b) y = (ex + 10)/2
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Function FamilyThe graph of the function
y = f(x – h) k x – h = 0, x = h
is the graph of the functiony = f(x)
shift h unit to the right and k unit up/down.The graph of the function
y = f(x + h) k x + h = 0, x = –h
is the graph of the functiony = f(x)
shift h unit to the left and k unit up/down.
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Logarithmic Function FamilyThe graph of the logarithmic function has the following characterisitcs:
y = logb(x − h) + k
1.) The line x = h is a vertical asymptote.
2.) The domain is x > h, and the range is all real numbers.
3.) If b > 1, the graph moves up to the right. If 0 < b < 1, the graph moves down to the right.
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Example: Graph the function, state domain and range.
a) y = log1/2 (x + 4) + 2 b) y = log3(x – 2) – 1
1- 4
1 2
0
0
D: x > -4, R: all real numbers D: x > 2, R: all real numbers
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NOTICE
Natural logs (ln) will be graphed in the same way. Just pick points from the table on your graphing
calculator.
Be careful! There is a difference between:
Vertical shift
Horizontal shift
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pg. 491 #49-52, 58-63, 65-67
HOMEWORK