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
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
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
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
LOGS TO THE RESCUE
O S W E G OINTRODUCING…
O S W E G Ohich
xponent
oes
n
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.”
EVALUATE THE EXPRESSIONS
4
2
-2
-3
6Which Exponent
Goes On…
SPECIAL LOGARITHM VALUES
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…?
REWRITING LOGARITHMIC EQUATIONS
Logarithmic Form Exponential Form
COMMON NOTATION
EVALUATING COMMON & NATURAL LOGS
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
Practice: Evaluate the common and natural logarithm.
a) ln0.25
b) log3.8
c) ln3
d) lne2007
0.845
-1.386
0.580
1
pg. 490 #16-19, 24-30, 36, 37
HOMEWORK
LOGARITHMIC FUNCTIONSSection 8.4 (Day 2)
RULE!
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
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?
WHAT DOES THIS MEAN?
This means that they offset each other, or they “undo” each other.
These two functions are inverses to each other.
USING INVERSES: SIMPLIFY THE EXPRESSION
x
x
x
x
USING INVERSES: SIMPLIFY THE EXPRESSION
x
2x
2x
3x
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…
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.
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.
Examples: Find the inverse of the function
a) y = log8x
b) y = ln(x – 3)
Answers:a) y = 8x
b) y = ex + 3
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
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.
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.
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
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
pg. 491 #49-52, 58-63, 65-67
HOMEWORK