Post on 08-Sep-2018
Math-3 Lesson 8-5
Review Unit 5 (part 1)(Logarithms)
Vocabulary: The “Initial” Value
xxf )3(7)(
The initial value of the function is the coefficient
of the exponential.
xxg )5(3)(
What is the initial value of the following functions ?
xxg 7)(
How much money will be in a savings account
5 years after $300 was deposited if the
interest rate is 6%?
trAtA )1()( 0
Amount of $$
In the account
Initial
deposit
Annual
interest
rate
Years after
the deposit
5)06.01(300)5( A
47.401$)5( A
How much money will there be in an account after
5 years if it earns 6% interest compounded
quarterly if the initial deposit is $300?
kt
krAtA )1()( 0
Amount of $$
In the accountInitial
deposit
Annual
interest rateYears after
the deposit
# of times per year
interest is paid
)5(4)4
06.01(300)5( A
06.404$)5( A
Continuous compounding When they pay the
interest continuously.
rteAtA 0)(
Amount of $$
In the account
Initial
deposit
Annual
interest rate
Years after
the deposit
You can find this
on your calculator
How much money will there be in an account after 5 years
if it earns 6% interest compounded continuously if the initial
deposit is $300?
rteAtA 0)(
Amount of $$
In the account
Initial
deposit
Annual
interest rate
Years after
the deposit
)5(06.0300)5( eA
96.404$)5( A
Transforming Exponential FunctionsDescribe how to transform the graph of:
Into the graph of :
xxf 2)(
42*3)( 2 xxg VSF = 3, right 2, down 4
What is the:
a) Initial value
b) y-intercept
c) x-intercept
d) Horizontal asymptote
e) Domain
f) Range.
a) 3
b) 413
c) 2.415
d) y = -4
e) (-∞, ∞)
f) (-4, ∞)
Exponential Growth and Decay
xabxf )(
For what range of values of ‘b’
will result in exponential growth ?
For what range of values of ‘b’
will result in exponential decay ? 0 < b < 1
b > 1
Your turn: What is the equation?
diff in x
x -2 -1 0 1 2 3 4 5y 2 6 18 54 162 486
diff in y
+1 +1 +1 +1
*3 *3 *3*3
1. ‘f(x)’ is exponential xabxf )(
2. “initial value” = 2 2a xbxf 2)(
3. “growth factor” = 3 3b xxf )3(2)(
4. Check it: f(3) = ?3)3(2)3( f 27*2 54
54)3( f
Comparing Rates of Growth: Which function grows the fastest?
xxf 3)( 2)( xxg
3)( xxh
4)( xxk
]000,1 ,10[y
]100,10[y ]100,10[y ]100,10[y ]100,10[y
]100,10[y ]100,10[y ]000,10 ,10[y
xxf 3)( 2)( xxg 3)( xxh
4)( xxk
xxf 3)(
window
4)( xxk
xxf 3)( 4)( xxk
xxf 3)(
Domain and Range
• The domain of f is the range of f -1
• The range of f is the domain of f -1
xxf log)(
Exponential Function
xxf 10)(
Domain = ?
Range = ?
Domain = ?
Range = ?
(-∞, ∞)
(0, ∞)
Logarithm
Function
Inverse
Functions
(0, ∞)
(-∞, ∞)
Horizontal asymptote = ?
y = 0Vertical asymptote = ?
x = 0
Transformations of the Log Function
xxf log)( )1log()( xxgRight 1 translation
Domain = ?
Range = ?
(1, ∞)
(-∞, ∞)
vertical asymptote is the
left/right shift
x = 0 asymptote = ? x = 1
Transformations of the Log Function
xxf log)(
1)2log(3)( xxgReflected x-axis
VSF = 3
Right 2 translation
Up 1 translation
Domain = ?
Range = ?
(2, ∞)
(-∞, ∞)
asymptote = ? x = 2
This is NOT an exponential
(has a vertical asymptote
NOT a horizontal asymptote).
What is a logarithm?
A logarithm is another way of writing an exponent.
x8log 282 x
Both of these equations are saying the same
thing:
“2 raised to what power is 8?”
x is the exponent Log = exponent
base = logarand
Convert to exponential form
x100log 10
5)27(log3 x
6)42(log3 9 x
10010 x
2735 x
429)
36(
x
exponent
base
exponentlog (logarand) =
4292 x
What does it mean?8log
x8log 810 x
010 x10
1
110
108
8.0x 9.0x
Find on your calculator.8log 903.08log
10 raise to what exponent equals 8?
Finding the Inverse
xxf )5(2)(
?)(1 xf
yx )5(2
yx)5(
2 Base: 5
“A log is an exponent”
2log5
xy
2log)( 5
1 xxf
Finding the Inverse
2)3()( 1 xxf
?)(1 xf
Base: 3
“A log is an exponent”
2log1 3 xy
12log)( 3
1 xxf
2)3( 1 yx
1)3(2 yx
“isolate” the exponential”
12log3 xy
Properties of Logarithms: Log of a Product.
SRRS bbb loglog)(log
5log3log15log 222
Log of a “product”
log of a product = sum of the logs of the factors.
25xx
Logarithm: another way of writing the exponent
Analogous to:
Product of powers
Add the exponents25 x
xy3log yx 33 loglog
“Expanding the Product”
45log 3
5log3log3log 333
5*3*345
5log3log2 33
Properties of Logarithms
3log4 2
52log
2log5
Log of a Power
c
b Rlog
4
2 3log 32log
cRc blog
25x Logarithm: another way of
writing the exponent
2*5x10x
Gotcha’
Which one?y3log5
53log y5log3log y
y3log5 53log y 553log y 53log y
Exponent ‘5’ is applied to ‘y’? or both ‘3’ and ‘y’ ?
Properties of Logarithms Log of a Quotient
SRS
Rbbb logloglog
2
5log 3
2log5log 33 “expand the quotient”
3ln8ln “condense the quotient”
3
8ln
“Negative Log” denominator of the logarand
Expand the Logarithm
Factors of numerator become
positive logs
53
2log
y
x
yx log53log2log
yx log53loglog2log
53log2log yx
Factors of denominator
become negative logs
Change of base formula: can convert any base to any other base
?9log 2 2log
9log
301.0
9542.0 17.3
?9log 2
2ln
9ln
69315.0
19722.2 17.3
BUT, more useful to convert to either (1) base 10, or (2) base ‘e’