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Math‐3
Unit 5 Review(Logs and Exponentials and their
Applications)
What is the equation that passes through the data?
x y
1
2
3
4
5
0.250.5124
xABxf )(31 AB42 AB
3
4
12
ABAB
B23)2(1 A
81
A
xxf )2(81)(
xxf )2(21)( 3
xxf )2(2)( 332)( xxf
The graph of can be obtainedfrom by reflecting it acrossthe y-axis.
Transforming Exponential Functions
xxf 2)(xxf 2)(
xxf 2)(xxf 2)( xxxf
212)(
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:Identify the following as either “growth” or “decay”.
xxf 2)(
xxf )3()(
xxf 31)(
xxf 234)(
dabxf cx ))(1()1()(If negative:
Reflect across x-axis
Initial value:Crosses y-axis at
y = a + dGrowth factor:
If negative:Reflect across y-axis
Horizontal shift
vertical shift and horizontal Asymptote
xxf 2)3(4)( )2(1)3(4 x
Finding the Inverse: exchange the locations of ‘x’ and ‘y’ in the equation then solve for ‘y’.
2)2()( xxf2)2( xy2)2( yx
2)2( yx
2 yx
yx 2
xy 2
xxf log)(
Exponential Function
xxf 10)( Domain = ?Range = ?
Domain = ?Range = ?
(-∞, ∞)(0, ∞)
LogarithmFunction
InverseFunctions
(0, ∞)(-∞, ∞)
Horizontal asymptote = ? y = 0 Vertical
asymptote = ?x = 0
Transformations of the Log Functionxxf log)(
1)2log(3)( xxgReflected x-axisVSF = 3Right 2 translationUp 1 translationDomain = ?Range = ?
(2, ∞)(-∞, ∞)
asymptote = ? x = 2NOT exponential (has a vertical asymptote NOT a horizontal asymptote.
Exponential LogarithmForm Form
x25log5
x64log4
x81log9
x1000log10
255 x
644 x
819 x
100010 x
Log =
Your Turn:
What is the base?
x5lnx20log
x8log2
Finding the Inverse
2)3()( 1 xxf
?)(1 xf
Base: 3“A log is an exponent”
2log1 3 xy
12log)( 31 xxf
2)3( 1 yx1)3(2 yx
“isolate” the exponential”
12log3 xy
xy3log yx 33 loglog
Log of a Product Property
45log3
5log3log3log 333
5*3*345
5log3log2 33
“expand the product”
xlog5log “Condense the Product” x5log
Log of a Quotient Property
SRSR
bbb logloglog
25log3 2log5log 33 “expand the quotient”
3ln8ln “condense the quotient”38ln
3log4 2
2log5
Log of a Power Property
cb Rlog
42 3log
32log
c Rc blog
52log
“expand power”
“condense”
Your turn:
Solve: 353 xx
“Isolate the radical”
“Undo the radical”
22)3(53 xx square both sides
9653 2 xxx Get into standard form!!!!!
2 ,7x
1490 2 xx)2)(7(0 xx
3)7(5)7(3
416 3)2(5)2(3
11
xx 41312 77
In some books, this is a property.xx 41312
If the base of exponentials are the same then the exponents equal each other.
1316 x
126 x
+4x +4x
-1 -1
2x
Solving using “convert to same base”
“convert to same base”12 279 xx
1322 33
xx
)1(32*2 33 xx Power of a power Exponent Property
334 33 xx
334 xx
3x
If you don’t recognize: “convert to same base”Use Log of a Power Property12 279 xx
12 27ln9ln xx
27ln)1(9ln2 xx
3x
127ln
9ln2 xx
127ln9ln2 xx
1)6.0(2 xx
1322
xx
334 xx
1275 xx
127ln5ln xx
85.0x
7ln)12(5ln xx
5ln7ln)12( xx
÷ ln 5 ÷ ln 5
21.142.2 xx
xx 42.221.1
x42.121.1
Cannot rewrite the exponentials with the same bases.
)21.1)(12( xx
+1.21 +1.21
-x -x
÷ 1.42 ÷ 1.42
Solve using “undo the exponential”
“Isolate the exponential”
“Undo the exponential”
753 12 x
-5 -5
Change of baseformula
23 12 x
122log3 x
123ln2ln
x
+1 +1÷2 ÷2
815.0x
1263093.0 xx263093.1
Some would say this is a property.“same based log equals same based log”)5(log)74(log 55 xx
Logarands equal each other.4x - 7 = x + 5
3x = 12
x = 4
Subtract ‘x’ from both sides.
Divide both sides by 3
Plug back in to check!)54(log)74*4(log 55
9log9log 55 Checks
Solving Logarithmic Equations“Isolate the logarithm”3)15(log4 x “convert to exponential”
1543 x
5x - 1 = 64
655 xx = 13 Plug back in to check!
Checks
3)113*5(log4
364log4
Solving Logs requiring condensing the product.
“Isolate the logarithm”2)5log(2log xx“undo the logarithm”
2)5(2log 1010 10 xx
“condense the product”
2x(x - 5) = 100
Inverse of log base 10 is exponent base 10
Quadratic put in standard form
Divide both sides by ‘2’factor
2)5(2log xx
0100102 2 xx05052 xx0)5)(10( xx
x = 10, -5 Zero factor property
x = 10x ≠ -5
We can rewrite the base of any exponential as a power of ‘e’.
693.0k
xy 2
2lnk2ke
xkey kxey
xey 693.0
xy 4 xe 386.1xy 1.1
xy 01.1xy 85.0xy 25.0
xe 095.0xe 010.0
xe 163.0xe 386.1
How can you tell if a base ‘e’ exponential is growth or decay?
kxey Growth: k > 0
Decay: k < 0
Newton’s Law of CoolingA high temperature item will cool off in a lower temperature medium in which it is placed. This cooling off process can be modeled by the following equation.
ktmom eTTTtT )()(
Temperature (as a function of time)
Temperature ofthe medium
Initial Tempof the object
Constant, determinedby the heat transfercharacteristics of the materi
Time
3070)( 11.0 tetT
A cup of hot water is taken out of the microwave oven. Its initial temperature is 100 C.It is placed on the counter in a room whose temperature is30 C. In 5 minutes it has cooledto 72 C. When will it reach 40 C.
1. Draw a graph that shows temperature as a function of time.
2. What is the equation of the graph? (use the following equation). kABtT t )( 3. Convert base “B” to base “e”
then rewrite your equation.
30)9.0(70)( ttT4. T(t) = 40 C. Substitute and solve for ‘t’. min 7.17t
The front row of a rock concert has a sound intensity of
What is the sound level in decibels on the front row of the rock concert?
21 er watts/met10I
1210log10)(
IIL
12
1
1010log10
L 1110log10 10log110dB 110
The loudness of an ambulance was measured to be 120 dB.
What is the sound intensity? (in w/m^2)
1210log10120
I1210
log12 I 12
12
1010
I
01212 101010 I
AciditypH = - log [H+]
The hydrogen-ion concentration of a solution is mole/li 107.5 11
What is the pH of the solution?
pH = - log [ ] 11107.5
pH = 10.3
You deposit $100 money into an account that pays 3.5% interest per year. The interest is “compounded” monthly. How much money will be in the account at the end of the 5th year?
kt
krAtA )1()( 0
)5(12)12035.01(100)5( A
09.119$)5( A
A bank compounds interest continuously. The annual interest rate is 5.5%. How long would it take for the money in the account to triple?
rteAtA 0)( teAA 055.0
003 te 055.03
t055.03ln yrs 97.19t
The “half life” of Carbon-14 (a radioactive isotope of carbon), is 5730 years. Calculate the decay rate for carbon-14.
kteAtA 0)( )5730(
005.0 keAA )(57305.0 ke
k57305.0ln 00012.0k
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