Switch x and y. 3 -1 = 1 / 3 3 0 = 1 3 1 = 3 3 2 = 9.
-
Upload
virginia-copeland -
Category
Documents
-
view
235 -
download
0
Transcript of Switch x and y. 3 -1 = 1 / 3 3 0 = 1 3 1 = 3 3 2 = 9.
b > 0 and b = 1
V.A. at x = 0
Common Point at ( 1, 0 )
Domain: Range: , ,0
b > 1: increase; 0 < b < 1: decrease
b > 0 and b = 1
H.A. at y = 0
Common Point at ( 0, 1 )
Domain: Range: , ,0
b > 1: increase; 0 < b < 1: decrease
SWITCH X & Y CONCEPTS!
Negative, flip over the x-axis. 0 < a < 1, Vertical Shrink and a > 1, Vertical Stretch.
Negative, flip over y-axis. 0 < b < 1, Horizontal Stretch and b > 1, Horizontal Shrink.
Solve for x. This is the Horizontal shift left or right.
This is the Vertical shift up or down.
Translate
Graph
The minus 2 is inside the function and solve for x. x = +2, shift to the right 2 units.
x = 0
xxf 3log
2log3 xxf
x = 2
Translate
Graph
The minus 5 is outside the function and shift down 5 units.
xxf 3log
5log3 xxf
x = 0
Translate
Graph
The negative on the x will flip the graph over the y-axis and solve 1 – x = 0 to determine how we shift horizontally. x = 1
x = 0
xxf 3log
xxf 1log3
x = 1
Translate
Graph
The plus 2 is inside the function and solve for x. x = -2, shift to the left 2 units. The minus 1 will shift down 1 unit.
x = 0
xxf 3log
12log3 xxf
x = -2
Translate
Graph
The negative on the x will flip over the y-axis. The negative in front of the log will flip the graph over the x-axis.
x = 0
xxf 3log
xxf 3log
Base e is the value 2.718281828…...
15log3 10 xyReplace f(x) with y.
Notice that there is no base number listed. Put in a 10.
154 2 xySwitch x and y. 15log3 10 yx
Solve for y. 5log31 10 yx
5log3
110
y
x
Convert to exponential.510 3
1
yx
yx
510 3
1 510 3
11
x
xf
154 2 yx
2541 yx
254
1 yx
24
1log5
yx
Convert to logarithm.
yx
24
1log5
24
1log5
1
xxf
Isolate the log function or exponential function.
The log function is isolated.Covert to exponential.
7432 x
749 x
x416
x4
642 x
8x
8xHas to be +8 because the base must be positive.
The exponential function is NOT isolated.Divide by 3.
102 xeCovert to logarithmic.
x210ln
xe 210log
Should be written with ln.
x
2
10ln
The base of 2 on the log and inside the ( )’s cancel out.
3 112 2log
3 11
The base of e and the ln in the exponent cancel out.
13lne
13
The 2 log base 2 can be condensed into one log.
16log3
48log
3log48log
22
22
42log 42
16 = 2*2*2*2
Set the exponent = x.
xe
9log 2
Use the base change formula for both logs.
8log
9log
3log
8log
The 2 log base 6 can be condensed into one log.
36log49log
4log9log
66
66
26log 26
36 = 6*6
Convert to exponential.
92 x
e
92 xeConvert to natural log and solve for x.
2
1
9ln9ln2
1
9ln2
x
x
33ln9ln ee
3log
9log
23log
9log2
3
3
Place the value of x back as an exponent on e.
The goal here is to factor the argument of the inside the log function to a 2, 3, or 5.
322log b
3 4
2 2
3log2log2log bbb 0986.16931.06931.0
4848.2
Convert the decimal to a fraction.
10
27logb
3 3 3
5 2
52
3log
3
bProduct RuleQuotient Rule
5log2log3log 3bbb
Power Rule 5log2log3log3 bbb
6094.16931.00986.13 9933.0
2 3 5 b
bbbbb log5log3log2log
16094.10986.16931.0
4011.4
2
1
6logb 6log2
1b
Power Rule
32log2
1 b 3log2log
2
1bb
0986.16931.02
1 89585.07917.1
2
1
Change-of-Base formula is
bM
b
MMb ln
ln
log
loglog
13log
39log39log13
Conversion Check
21log
5log5log21
11log
8log8log
11
13ln
39ln39log13
21ln
5ln5log21
11ln
8ln8log
11
We will use the Product, Quotient, and Power Rule to expand. Simplify, if possible.
Product Rule
yx 22
22 loglog8log
Power Rule
yx 222 loglog28log
Breakdown the 8 = 23
yx 223
2 loglog22log
yx 22 loglog23
Notice that there is a log for every factor. This is true whether the factors are top or bottom. Remember that the factors from the bottom are always minus the log.
Count the factors.1 2
3 4
A log for each.
33333 1log9loglog4log xy
Plus logs from the top. Minus logs from the bottom.
Simplify individually with Power Rule.
333
2
1
33 1log9loglog4log
xy
1log39loglog2
14log 3333 xy
One more.
1log33loglog2
14log 3
2333 xy
1log32log2
14log 333 xy
We will use the Product, Quotient, and Power Rule to condense. Simplify, if possible.
We must have a log in every term!The 2 is the answer from a simplified log. This means that is was the exponent on the base inside of a log5(5).
11log2
1log35loglog4 5555 yx
2
11log2
1log35loglog4 55
255 yx
Move all coefficients back inside as powers.
2
1
53
52
54
5 11loglog5loglog yx
Simplify the insides, if possible.
11loglog25loglog 53
554
5 yx
Optional.
3
4
5 25
11log
y
x
Plus logs to the top. Minus logs to the bottom.
We have a log in every term!
1
1log
2
3
3 x
x
top bottom
Simplify the inside…factor.
11
11log
2
3 xx
xxx
Perfect Cube
Diff. of Squares
1
1log
2
3 x
xx
One-to-one Property Conversion Rule
1
33
213
x
x
x
1332 x
We have a single log, convert to exponential.
139 x
x
x
3
10
310
Solve
Not yet One-to-one Property.Power Rule to move coefficients. 3
52
5 4loglog x
Now, One-to-one Property.Cancel logs.
32 4x
642 x
642 x
8xThe Domain must be > 0…no negative 8 as a solution.
8x
Condense to one log on the left side. Use the Power and Product Rule.
54log1log3 22 x
54log1log 23
2 x
514log 32 x
Now that there is one log, convert to exponential
35 142 x 32
81 3 x
21x
3x
3 3
2 2log 7 log 8 1x x Solve
24 4log 9 log 3 3x x
Condense to one log on the left side. Use the Product Rule.
187log2 xx
Now that there is one log, convert to exponential
8721 xx
base exponent
FOIL
56152 2 xxSet = 0.
054152 xxFactor
9,6
096
x
xx
Now check answers…-9 doesn’t work. -9 creates negative values inside the log.
Condense to one log on the left side. Use the Quotient Rule.
3
3
9log
2
4
x
x
Factor and simplify the inside.
3
3
33log4
x
xx
33log4 xNow that there is one log, convert to exponential
343 x 64
67x
Now we isolate the exponential expression.
153 x Convert to a log.
x15log3
Our answer for x is considered an exact value. You may be asked to convert to a rounded decimal answer.Base Change Formula.
This is factorable. Notice the first power is double that of the second power.
122 uu
34 uu 03242 xx
Set = 0 and solve for x.
042 x 032 x
42 x 32 x
Convert to logs for both. x 4log2 x3log2
No negatives in log functions or exponentials can’t = negatives
This is NOT factorable. Notice the first power is NOT double that of the second power.
FORCE IT! AARRRRGGGGG!Use your exponential rules. When you multiply like bases, we add the exponents. So manipulate the middle term.
xxx 33333 11
043332 xx 14432 uuuu
01343 xxSet = 0 and solve for x.
043 x 013 x
43 x 13 xNot Possible
Convert to a log.
x1log3
0x
UGLY! Bases are different.Authors way.
The author wants you to take either the common log or natural log of both sides.
xx 211 5ln2ln Power Rule
5ln212ln1 xx Dist. Prop.
5ln25ln2ln2ln
5ln25ln12ln12ln
xx
xx
Move all the terms with an x to the left side. 5ln2 x 5ln2 x
5ln2ln5ln22ln xx 2ln 2ln
2ln5ln5ln22ln xx
Non- x terms to the right side.
Factor out x as GCF.
2ln5ln5ln22ln x Isolate x.
5ln22ln 5ln22ln
5ln22ln
2ln5ln
x Power Rule
25ln2ln
2ln5ln
x Product &Quotient
Rule
252ln25
ln
x
50ln
5.2lnx
Base Change
5.2log50x
Mr. Fitz’s way.Pick a base and convert it to a log of that base.
xx 212log 15
I will go base 5.
Power Rule
xx 212log1 5 Dist. Prop. xx 212log2log 55 Move all the
terms with an x to the left side. 12log22log 55 xx
x2x2
Non- x terms to the right side.
2log5 2log5 2log122log 55 xx
Factor out x as GCF. 2log122log 55 x
Isolate x. 22log5 22log5
22log
2log1
5
5
x
2
55
51
5
5log2log
2log5log
x
25
5
52log25
log
x
5.2log50log
5.2log50
5
5 x