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log 16 ?log 16 4
? 4
16 2?
2 2
? 4
Exponential Form
log 16 π₯
16 2 2 2
π₯ 4
Set Log arbitrarily π₯
Exponential FormChange of Base
Same Base: Make exponents equal to each other
Log Form β> Exponential Form and Solve for π₯
β²π ππ π‘βπ π‘βπππ π¦ππ’ πππ πππππππ
β²πβ² ππ π‘βπ πππ π.
β²π ππ π‘βπ πππ π€ππ/ππ₯ππππππ‘
πΏππ πΉπππ
πΈπ₯ππππππ‘πππ πΉπππ
log 9 ?log 9 2
? 2
Think: What power do you have to raise 3 to, to equal 9?
log 8 ?log 8 3
? 3
Switching from Log Form to Exponential Form:
Remember: The base of the log is the base of the exponent.
The exponent is the Answer.The thing you are Logging equals the Base to the other side.
The Definition of a Logarithm:
Log Form
8 equals 2 to what power?
16 equals 2 to what power?
log 16 4
log1
27β―β―β― π₯
1
27β―β―β― 3
1
3β―β― 3
3 3
π₯ 3
logβ―β―
16 π₯
1612β―β―
2 2 2 2 4 π₯ π₯ 4
log 16π π₯ 16π 2π 2π 2π
π₯ 4
Exponential FormChange of BaseExponent LawsSolve
Exponential Form
Change of BaseExponent Laws
Exponential FormChange of Base
C12 β 8.1 β log π ? Definition Notes
Log Page 1
log π₯ 2 π₯ 6
π₯ 36
The base of the log is the base of the exponent
log π₯ 2 π₯ 5
π₯1
5β―β―
π₯1
25β―β―β―
log π₯ 5 2 π₯ 5 2 π₯ 4 5
π₯ 9
πππ 2 1 2 π₯ 3 2 π₯ 3 π₯ 5
log 64 3 64 π₯ 4 π₯
π₯ 4
log 2732β―β―
27 π₯β―β―
27β―β― π₯β―β―β―β―
27β―β― π₯
27β―β―β―
π₯
π₯ 9
log 125 π₯ 125 5 5 5 π₯ 3
Exponential FormChange of Base
Same Base: Make exponents equal to each other
log π₯ 3 π₯ 4
π₯ 64
log 32 5 32 π₯ 2 π₯
2β―β―
π₯β―β―β―
π₯ 5
Exponential Form
Solve
Exponential FormChange of BaseFifth Root Both Sides
Solve
Log Form β> Exponential Form and Solve for π₯
Exponent Laws
Solve
log π₯12β―β―
π₯ 9β―β―
π₯ β9β―β―
π₯ 3
Exponential FormChange of Base
Solve
Take both/sides to reciprocal exponent
log 16 π₯ 2 16 2 2 2 4 π₯ 2
π₯ 2
log 16 π₯ 2log 16 2 π₯ 4 2 π₯
π₯ 2
log 16 π₯ 16 2 2 2 π₯ 4
log 16 4OR Do AlgebraFirst!
C12 β 8.1 β log π₯ π, log π π, log π π₯ πππ‘ππ
Log Page 2
log # π’ππlog # π’πππππ 0 π’ππ
πππ 3 π’ππ
log 5π₯ π₯12β―β―
5π₯ π₯ 36β―β―
5π₯ π₯ 6 π₯ 5π₯ 6 0 π₯ 2 π₯ 3 0
π₯ 2 , π₯ 3
5π₯ π₯ 0π₯ 5 π₯ 0
0 π₯ 5
log π₯ 112β―β―
π₯ 1 9β―β―
π₯ 1 3 π₯ 4 0 π₯ 2 π₯ 2 0
π₯ 2 , π₯ 2
π₯ 1 0π₯ 1 π₯ 1 0
π₯ 1 , π₯ 1
log π₯ 2 π₯ 3 π₯ 9
π₯β―β―β―
β9β―β―
π₯ 3
π₯ 3 , π₯ 3
2 log π₯ 2 log π₯ 1 π₯ 1 π₯ 9
π₯β―β―β―
β9β―β―
π₯ 3
π₯ 3 , π₯ 3
π₯ 0 π₯ 0π₯ 0, π₯ 0
π₯ 0
log 3 π₯ 3 3 π₯ 2 3 π₯ 8
π₯ 5
3 π₯ 0 π₯ 3
π₯ 3
πππ 2 2 2 π₯ 3 2 π₯ 3 π₯ 3 2 π₯ 6π₯ 9 0 π₯ 6π₯ 7 0 π₯ 7 π₯ 1
π₯ 7 , π₯ 1
π₯ 3 0
π₯ 3
log # π’ππ
ππππ₯
π₯ 0
log #
π₯ 0 , π₯ 1
log π₯ 5 2 π₯ 5 2 π₯ 4 5
π₯ 9
log π₯ 2 π₯ 2
π₯ 4
State Restrictions and Solve
State Restrictions
π₯ 0 π₯ 5 0
π₯ 5
Domain: Set the thing you are logging to greater than or equal to zero, then solve.
π₯ 3 1
π₯ 4
Set the base of the log 0 πππ 1 πππ π πππ£π.
C12 β 8.2 β Log Restrictions Notes
Log Page 3
log 5 π₯ 5 5
π₯ 4
Change of BaseBring Exponent down in frontLog Rules
Solve
log 5 1
πππ 625 π₯ log 5 π₯ 4log 5 π₯ 4 1 π₯
π₯ 4
5 to what power is 5
Bring Exponent down in front and Vice Versa
3πππ42 3πππ4 6πππ4
πππ52πππ5
ππππ₯2ππππ₯ 3πππ4
πππ4πππ4
OR
πππβπ₯β―β―
ππππ₯β―β―
12β―β―log π₯
ππππ₯
2β―β―β―β―
log12β―β―
πππ21πππ2
πππ2
πππ 625 π₯ 625 5 5 5
π₯ 4
C12 β 8.3 β log π πππππ Notes
Log Page 4
log 16πππ 4
β―β―β―β―β―β―
log 16 2 Change of Base
log 27πππ27πππ3
β―β―β―β―β―
πππ3πππ3
β―β―β―β―β―
3πππ3πππ3
β―β―β―β―β― 3
1
log 2β―β―β―β―β―
1
πππ2πππ8β―β―β―β―
β―β―β―β―β―β―
1πππ8πππ2β―β―β―β―
πππ8πππ2β―β―β―β―
log 8 3
log 16log 16log 8
β―β―β―β―β―β―43β―β―
Change of Base
Change of Base
Change of Base
27 3
Exponential Form
16 4
16 28 2
8 2
Change of Base
log 4log 4log 2β―β―β―β―β―
21β―β― 2
4 2
2 2
Exponent down in front
πππ16πππ2
β―β―β―β―β―
log 16 416 2
Change of Base
Choose the Base you want! Think about it.
C12 β 8.3 β Change of Base Notes
Log Page 5
log 4 log 8 log 4 8 log 32 5
32 2
Exponential FormAddβMultiply
log 3 log 9 πππ 3 9 log 27 3 AddβMultiply
27 31 2 3
2 3 5
log 27 log 3 log273
β―β―β― log 9 2
3 1 2
SubtractβDivide
πππ64 πππ4 16 πππ 4 πππ16 Separate into an addition of logs
πππ5 log102
β―β―β― πππ10 πππ2Separate into a subtraction of logs
πππ1 πππ5 πππ7 πππ1 5 7 log 35 AddβMultiply
πππ5 πππ2 πππ10 πππ5
2 10β―β―β―β―β―β― log
14β―β―
πππ5 πππ2 πππ10 πππ5 10
2β―β―β―β―β―β― πππ25
πππ4 πππ20 πππ10 πππ4 20
10β―β―β―β―β―β― πππ 8 Positives on top, Negatives on Bottom
C12 β 8.4 β log π log π log ππ πππ π πππ π πππ β―β―
Log Page 6
AddβMultiply
SubtractβDivideβFactorβSimplify
log 3 log π₯ 1 πππ3 π₯ 1 log 3π₯ 3
log π₯ 2 log π₯ 1 log π₯ 2 π₯ 1 log π₯ π₯ 2 AddβMultiply
log π₯ 1 log π₯ 1 logπ₯ 1 π₯ 1
β―β―β―β―β―β― logπ₯ 1 π₯ 1
π₯ 1β―β―β―β―β―β―β―β―β―β―β―β―β― log π₯ 1
ππππ₯ log π₯ log π₯ π₯ log π₯AddβMultiply
log π₯ log π₯ logπ₯π₯β―β―β― ππππ₯
SubtractβDivideβSimplify
C12 β 8.4 β log π log π log ππ πππ π πππ π πππ β―β―
Log Page 7
πππ5 πππ2 πππ5 log 2 0.2104
πππ62πππ6 1.5563 Bring your exponent down in front.
πππ5 π₯ 2 πππ5 π₯πππ5 2πππ5
ππππ₯π¦ππππ₯ ππππ¦ππππ₯ 2ππππ¦
ππππ₯ π¦ππππ₯ ππππ¦2ππππ₯ 2ππππ¦
log π₯π¦2ππππ₯π¦2 ππππ₯ ππππ¦2ππππ₯ 2ππππ¦
The exponent only applies to the y value
You can bring this exponent down in front.Remember: If you separate into an addition you must distribute the 2.
3π₯πππ7 π₯πππ2π₯ 3πππ7 πππ2 πΊπΆπΉ π₯
π·ππ π‘ππππ’π‘π
πππ25 1.3979πππ5 1.39792πππ5 1.3979
25 5Change of Base If for example you know:
πππ5 π 2πππ52π
ππππ₯ ππππ₯ ππππ₯ ππππ₯
πππ2π₯ππππ₯
β―β―β―β―β―πππ2π₯ππππ₯
β―β―β―β―β―
ππππ₯ ππππ₯ ππππ₯ Cannot bring exponent down in front
Cannot Divide 2 logs
Cannot multiply 2 logs
2πππ5 2πππ5 Cannot distribute into a log
log π₯ 2 log π₯ 2 Cannot distribute
πππ8 0.9031 Calculatorπππ 7 1.4037 Math, Alpha, Math
2πππ53 2πππ5 6πππ5 4.1938
If there is a number in front multiply
C12 β 8.5 β Log Operation Notes
Log Page 8
log 8 log 8 log 64 38 2 64 4
Exponential Form
Take the base and the log to any exponent you like!
logβ―β―
4 logβ―β―
4 log 4 1 log 4 214β―β― 2
12β―β― 2
log 4 24 2
Bring your exponent down in front
1 2 2
ππππ π‘βπ πππ π πππ π‘βπ π‘βπππ π¦ππ’ πππ πππππππ π‘π ππ ππ₯ππππππ‘ π‘π πππ‘ ππππ πππ ππ π‘π π’π π log πππ€π
C12 β 8.5 β Operation log π πππ‘ππ
Log Page 9
4 2 πππ4 πππ2 πππ4 π₯πππ2
πππ4πππ2β―β―β―β― π₯
log 4 π₯ 4 2 2 2 π₯ 2
Log Both SidesBring Exponents Down In FrontDivide
Change of baseExponential FormChange of baseSolve
8 2 2 4 2
3 2 14 2
Divide FirstAdd/Subtract First
Deβlog Both sides
log 4 log π₯ 4 π₯
3 5 πππ3 πππ5 πππ3 π₯πππ5
πππ3πππ5β―β―β―β― π₯
log 3 π₯ π₯ 0.6826
Remember: You may only log both sides if SAMD is complete. Bedmas backwards.Remember: If you do log a product you must separate into an addition of logs.Remember: if you log a sum you must use bracketsRemember: You may only deβlog both sides if one log equals one log.
π΄ππππππππ πππ π€ππ
4 7 πππ4 πππ7 πππ4 π₯ 1 πππ7 πππ4 π₯πππ7 log 7πππ4 πππ7 π₯πππ7πππ4 πππ7
πππ7β―β―β―β―β―β―β―β―β―β―β― π₯
π₯πππ4 πππ7
πππ7β―β―β―β―β―β―β―β―β―β―β―
π₯ 0.29
Check Answer:
5 . 3
2 9 πππ2 πππ9 2π₯ 5 πππ2 π₯ 2 πππ9 2π₯πππ2 5πππ2 π₯πππ9 2πππ9 2π₯πππ2 π₯πππ9 2πππ9 5πππ2π₯ 2πππ2 πππ9 2πππ9 5πππ2
π₯2πππ9 5πππ22πππ2 πππ9
β―β―β―β―β―β―β―β―β―β―β―β―β―
DistributeCombine x's on one sideEverything else on other sideFactor out xDivide
πΊπΆπΉ π₯Divide
Before you log both sides!
8 3 πππ8 πππ3 πππ8 2π₯πππ3
πππ8πππ3β―β―β―β― 2π₯
log 8
2β―β―β―β―β― π₯
12β―β―log 8 π₯
x log 8β―β―
Bring Fraction In Front.Bring Coefficient Up
Or
8 2 2πππ8 log 2 2πππ8 πππ2 πππ2
Or divide log7 and minus 1
6 3 14 πππ6 3 πππ14 πππ6 πππ3 πππ14 πππ6 π₯πππ3 2π₯ 5 πππ14 πππ6 π₯πππ3 2π₯πππ14 5πππ142π₯πππ14 π₯πππ3 πππ6 5πππ14π₯ 2πππ14 πππ3 πππ6 5πππ14
π₯πππ6 πππ15
2πππ14 πππ3β―β―β―β―β―β―β―β―β―β―β―β―β―
π π’ππ 6 πππππ π π₯ log a ππππ₯log π₯ log π ππππ₯log π₯ log π
ππππβ―β―β―β―β―β―β―β―β―β―β―
ππππ₯log πβ―β―β―β―
log π₯ππππ₯ππππβ―β―β―β―
log π₯ log π₯ π₯ π₯
C12 β 8.6 β Log/Delog Both Sides Notes
Log Page 10
π¦ log π₯
π¦ 2
π₯ π¦
π₯ π¦
1 12β―β―
0 1
1 2
π₯ π¦
12β―β―
1
1 0
2 1
πΊπππβ:
π¦ log π₯
π¦ 2
π¦ log π₯ 2 1 πΊπππβ:
π¦ 2
π¦ 2
π₯ π¦
π¦ log π₯
π¦ log π₯ 2 1
πΏπππ‘ 2ππ 1
π·πππππ: π₯ 0
π₯ 2 0 π₯ 2
ππ΄: π₯ 0
π₯ 2 0 π₯ 2
C12 β 8.7 β Graph Log Notes
Log Page 11
π¦ 2
1,12β―β―
0,11,2
2,4
12β―β―, 1
1,02,1
4,2
y log π₯
Remember: Inverse: Switch x and y Remember: A diagonal reflection over the line π¦ π₯
π¦ 2 π₯ 2 ππππ₯ πππ2 ππππ₯ π¦πππ2
ππππ₯πππ2β―β―β―β― π¦
log π₯ π¦ π¦ log π₯π π₯ log π₯
y log π₯ π₯ log π¦ 2 π¦ π¦ 2π π₯ 2
π¦ 2 π₯ 2 π¦ log π₯π π₯ log π₯
π₯ π¦
0 π’ππ
12β―β―
1
1 0
2 1
2 4
π₯ π¦
1 12β―β―
0 1
1 2
2 4
ππ€ππ‘πβ π₯ πππ π¦Log Both SidesBring Exponents Down In FrontDivide
Change of baseMirrorInverse Function notation
ππ€ππ‘πβ π₯ πππ π¦Exponential to πππ Form
π΅πππ π‘βπ ππ‘βππ πππ¦!
π¦ 2 3 π₯ 2 3 π₯ 3 2 log π₯ 3 π¦ 1 πππ2
log π₯ 3
πππ2β―β―β―β―β―β―β―β―β― π¦ 1
log π₯ 3 π¦ 1log π₯ 3 1 π¦ π¦ log π₯ 3 1 π π₯ log π₯ 3
π¦ log π₯ 3 1 π₯ log π¦ 3 1 π₯ 1 log π¦ 3 2 π¦ 32 3 π¦ π¦ 2 3 π π₯ 2 3
πΌππ£πππ π πππππ
C12 β 8.7 β Inverse Log Graphs Notes
Log Page 12
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