log 16 ?log 16 4
? 4
16 2?
2 2
? 4
Exponential Form
log 16 ๐ฅ
16 2 2 2
๐ฅ 4
Set Log arbitrarily ๐ฅ
Exponential FormChange of Base
Same Base: Make exponents equal to each other
Log Form โ> Exponential Form and Solve for ๐ฅ
โฒ๐ ๐๐ ๐กโ๐ ๐กโ๐๐๐ ๐ฆ๐๐ข ๐๐๐ ๐๐๐๐๐๐๐
โฒ๐โฒ ๐๐ ๐กโ๐ ๐๐๐ ๐.
โฒ๐ ๐๐ ๐กโ๐ ๐๐๐ ๐ค๐๐/๐๐ฅ๐๐๐๐๐๐ก
๐ฟ๐๐ ๐น๐๐๐
๐ธ๐ฅ๐๐๐๐๐๐ก๐๐๐ ๐น๐๐๐
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
27โฏโฏโฏ ๐ฅ
1
27โฏโฏโฏ 3
1
3โฏโฏ 3
3 3
๐ฅ 3
logโฏโฏ
16 ๐ฅ
1612โฏโฏ
2 2 2 2 4 ๐ฅ ๐ฅ 4
log 16๐ ๐ฅ 16๐ 2๐ 2๐ 2๐
๐ฅ 4
Exponential FormChange of BaseExponent LawsSolve
Exponential Form
Change of BaseExponent Laws
Exponential FormChange of Base
C12 โ 8.1 โ log ๐ ? Definition Notes
Log Page 1
log ๐ฅ 2 ๐ฅ 6
๐ฅ 36
The base of the log is the base of the exponent
log ๐ฅ 2 ๐ฅ 5
๐ฅ1
5โฏโฏ
๐ฅ1
25โฏโฏโฏ
log ๐ฅ 5 2 ๐ฅ 5 2 ๐ฅ 4 5
๐ฅ 9
๐๐๐ 2 1 2 ๐ฅ 3 2 ๐ฅ 3 ๐ฅ 5
log 64 3 64 ๐ฅ 4 ๐ฅ
๐ฅ 4
log 2732โฏโฏ
27 ๐ฅโฏโฏ
27โฏโฏ ๐ฅโฏโฏโฏโฏ
27โฏโฏ ๐ฅ
27โฏโฏโฏ
๐ฅ
๐ฅ 9
log 125 ๐ฅ 125 5 5 5 ๐ฅ 3
Exponential FormChange of Base
Same Base: Make exponents equal to each other
log ๐ฅ 3 ๐ฅ 4
๐ฅ 64
log 32 5 32 ๐ฅ 2 ๐ฅ
2โฏโฏ
๐ฅโฏโฏโฏ
๐ฅ 5
Exponential Form
Solve
Exponential FormChange of BaseFifth Root Both Sides
Solve
Log Form โ> Exponential Form and Solve for ๐ฅ
Exponent Laws
Solve
log ๐ฅ12โฏโฏ
๐ฅ 9โฏโฏ
๐ฅ โ9โฏโฏ
๐ฅ 3
Exponential FormChange of Base
Solve
Take both/sides to reciprocal exponent
log 16 ๐ฅ 2 16 2 2 2 4 ๐ฅ 2
๐ฅ 2
log 16 ๐ฅ 2log 16 2 ๐ฅ 4 2 ๐ฅ
๐ฅ 2
log 16 ๐ฅ 16 2 2 2 ๐ฅ 4
log 16 4OR Do AlgebraFirst!
C12 โ 8.1 โ log ๐ฅ ๐, log ๐ ๐, log ๐ ๐ฅ ๐๐๐ก๐๐
Log Page 2
log # ๐ข๐๐log # ๐ข๐๐๐๐๐ 0 ๐ข๐๐
๐๐๐ 3 ๐ข๐๐
log 5๐ฅ ๐ฅ12โฏโฏ
5๐ฅ ๐ฅ 36โฏโฏ
5๐ฅ ๐ฅ 6 ๐ฅ 5๐ฅ 6 0 ๐ฅ 2 ๐ฅ 3 0
๐ฅ 2 , ๐ฅ 3
5๐ฅ ๐ฅ 0๐ฅ 5 ๐ฅ 0
0 ๐ฅ 5
log ๐ฅ 112โฏโฏ
๐ฅ 1 9โฏโฏ
๐ฅ 1 3 ๐ฅ 4 0 ๐ฅ 2 ๐ฅ 2 0
๐ฅ 2 , ๐ฅ 2
๐ฅ 1 0๐ฅ 1 ๐ฅ 1 0
๐ฅ 1 , ๐ฅ 1
log ๐ฅ 2 ๐ฅ 3 ๐ฅ 9
๐ฅโฏโฏโฏ
โ9โฏโฏ
๐ฅ 3
๐ฅ 3 , ๐ฅ 3
2 log ๐ฅ 2 log ๐ฅ 1 ๐ฅ 1 ๐ฅ 9
๐ฅโฏโฏโฏ
โ9โฏโฏ
๐ฅ 3
๐ฅ 3 , ๐ฅ 3
๐ฅ 0 ๐ฅ 0๐ฅ 0, ๐ฅ 0
๐ฅ 0
log 3 ๐ฅ 3 3 ๐ฅ 2 3 ๐ฅ 8
๐ฅ 5
3 ๐ฅ 0 ๐ฅ 3
๐ฅ 3
๐๐๐ 2 2 2 ๐ฅ 3 2 ๐ฅ 3 ๐ฅ 3 2 ๐ฅ 6๐ฅ 9 0 ๐ฅ 6๐ฅ 7 0 ๐ฅ 7 ๐ฅ 1
๐ฅ 7 , ๐ฅ 1
๐ฅ 3 0
๐ฅ 3
log # ๐ข๐๐
๐๐๐๐ฅ
๐ฅ 0
log #
๐ฅ 0 , ๐ฅ 1
log ๐ฅ 5 2 ๐ฅ 5 2 ๐ฅ 4 5
๐ฅ 9
log ๐ฅ 2 ๐ฅ 2
๐ฅ 4
State Restrictions and Solve
State Restrictions
๐ฅ 0 ๐ฅ 5 0
๐ฅ 5
Domain: Set the thing you are logging to greater than or equal to zero, then solve.
๐ฅ 3 1
๐ฅ 4
Set the base of the log 0 ๐๐๐ 1 ๐๐๐ ๐ ๐๐๐ฃ๐.
C12 โ 8.2 โ Log Restrictions Notes
Log Page 3
log 5 ๐ฅ 5 5
๐ฅ 4
Change of BaseBring Exponent down in frontLog Rules
Solve
log 5 1
๐๐๐ 625 ๐ฅ log 5 ๐ฅ 4log 5 ๐ฅ 4 1 ๐ฅ
๐ฅ 4
5 to what power is 5
Bring Exponent down in front and Vice Versa
3๐๐๐42 3๐๐๐4 6๐๐๐4
๐๐๐52๐๐๐5
๐๐๐๐ฅ2๐๐๐๐ฅ 3๐๐๐4
๐๐๐4๐๐๐4
OR
๐๐๐โ๐ฅโฏโฏ
๐๐๐๐ฅโฏโฏ
12โฏโฏlog ๐ฅ
๐๐๐๐ฅ
2โฏโฏโฏโฏ
log12โฏโฏ
๐๐๐21๐๐๐2
๐๐๐2
๐๐๐ 625 ๐ฅ 625 5 5 5
๐ฅ 4
C12 โ 8.3 โ log ๐ ๐๐๐๐๐ Notes
Log Page 4
log 16๐๐๐ 4
โฏโฏโฏโฏโฏโฏ
log 16 2 Change of Base
log 27๐๐๐27๐๐๐3
โฏโฏโฏโฏโฏ
๐๐๐3๐๐๐3
โฏโฏโฏโฏโฏ
3๐๐๐3๐๐๐3
โฏโฏโฏโฏโฏ 3
1
log 2โฏโฏโฏโฏโฏ
1
๐๐๐2๐๐๐8โฏโฏโฏโฏ
โฏโฏโฏโฏโฏโฏ
1๐๐๐8๐๐๐2โฏโฏโฏโฏ
๐๐๐8๐๐๐2โฏโฏโฏโฏ
log 8 3
log 16log 16log 8
โฏโฏโฏโฏโฏโฏ43โฏโฏ
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 2โฏโฏโฏโฏโฏ
21โฏโฏ 2
4 2
2 2
Exponent down in front
๐๐๐16๐๐๐2
โฏโฏโฏโฏโฏ
log 16 416 2
Change of Base
Choose the Base you want! Think about it.
C12 โ 8.3 โ Change of Base Notes
Log Page 5
log 4 log 8 log 4 8 log 32 5
32 2
Exponential FormAddโMultiply
log 3 log 9 ๐๐๐ 3 9 log 27 3 AddโMultiply
27 31 2 3
2 3 5
log 27 log 3 log273
โฏโฏโฏ log 9 2
3 1 2
SubtractโDivide
๐๐๐64 ๐๐๐4 16 ๐๐๐ 4 ๐๐๐16 Separate into an addition of logs
๐๐๐5 log102
โฏโฏโฏ ๐๐๐10 ๐๐๐2Separate into a subtraction of logs
๐๐๐1 ๐๐๐5 ๐๐๐7 ๐๐๐1 5 7 log 35 AddโMultiply
๐๐๐5 ๐๐๐2 ๐๐๐10 ๐๐๐5
2 10โฏโฏโฏโฏโฏโฏ log
14โฏโฏ
๐๐๐5 ๐๐๐2 ๐๐๐10 ๐๐๐5 10
2โฏโฏโฏโฏโฏโฏ ๐๐๐25
๐๐๐4 ๐๐๐20 ๐๐๐10 ๐๐๐4 20
10โฏโฏโฏโฏโฏโฏ ๐๐๐ 8 Positives on top, Negatives on Bottom
C12 โ 8.4 โ log ๐ log ๐ log ๐๐ ๐๐๐ ๐ ๐๐๐ ๐ ๐๐๐ โฏโฏ
Log Page 6
AddโMultiply
SubtractโDivideโFactorโSimplify
log 3 log ๐ฅ 1 ๐๐๐3 ๐ฅ 1 log 3๐ฅ 3
log ๐ฅ 2 log ๐ฅ 1 log ๐ฅ 2 ๐ฅ 1 log ๐ฅ ๐ฅ 2 AddโMultiply
log ๐ฅ 1 log ๐ฅ 1 log๐ฅ 1 ๐ฅ 1
โฏโฏโฏโฏโฏโฏ log๐ฅ 1 ๐ฅ 1
๐ฅ 1โฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏ log ๐ฅ 1
๐๐๐๐ฅ log ๐ฅ log ๐ฅ ๐ฅ log ๐ฅAddโMultiply
log ๐ฅ log ๐ฅ log๐ฅ๐ฅโฏโฏโฏ ๐๐๐๐ฅ
SubtractโDivideโSimplify
C12 โ 8.4 โ log ๐ log ๐ log ๐๐ ๐๐๐ ๐ ๐๐๐ ๐ ๐๐๐ โฏโฏ
Log Page 7
๐๐๐5 ๐๐๐2 ๐๐๐5 log 2 0.2104
๐๐๐62๐๐๐6 1.5563 Bring your exponent down in front.
๐๐๐5 ๐ฅ 2 ๐๐๐5 ๐ฅ๐๐๐5 2๐๐๐5
๐๐๐๐ฅ๐ฆ๐๐๐๐ฅ ๐๐๐๐ฆ๐๐๐๐ฅ 2๐๐๐๐ฆ
๐๐๐๐ฅ ๐ฆ๐๐๐๐ฅ ๐๐๐๐ฆ2๐๐๐๐ฅ 2๐๐๐๐ฆ
log ๐ฅ๐ฆ2๐๐๐๐ฅ๐ฆ2 ๐๐๐๐ฅ ๐๐๐๐ฆ2๐๐๐๐ฅ 2๐๐๐๐ฆ
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.
3๐ฅ๐๐๐7 ๐ฅ๐๐๐2๐ฅ 3๐๐๐7 ๐๐๐2 ๐บ๐ถ๐น ๐ฅ
๐ท๐๐ ๐ก๐๐๐๐ข๐ก๐
๐๐๐25 1.3979๐๐๐5 1.39792๐๐๐5 1.3979
25 5Change of Base If for example you know:
๐๐๐5 ๐ 2๐๐๐52๐
๐๐๐๐ฅ ๐๐๐๐ฅ ๐๐๐๐ฅ ๐๐๐๐ฅ
๐๐๐2๐ฅ๐๐๐๐ฅ
โฏโฏโฏโฏโฏ๐๐๐2๐ฅ๐๐๐๐ฅ
โฏโฏโฏโฏโฏ
๐๐๐๐ฅ ๐๐๐๐ฅ ๐๐๐๐ฅ Cannot bring exponent down in front
Cannot Divide 2 logs
Cannot multiply 2 logs
2๐๐๐5 2๐๐๐5 Cannot distribute into a log
log ๐ฅ 2 log ๐ฅ 2 Cannot distribute
๐๐๐8 0.9031 Calculator๐๐๐ 7 1.4037 Math, Alpha, Math
2๐๐๐53 2๐๐๐5 6๐๐๐5 4.1938
If there is a number in front multiply
C12 โ 8.5 โ 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!
logโฏโฏ
4 logโฏโฏ
4 log 4 1 log 4 214โฏโฏ 2
12โฏโฏ 2
log 4 24 2
Bring your exponent down in front
1 2 2
๐๐๐๐ ๐กโ๐ ๐๐๐ ๐ ๐๐๐ ๐กโ๐ ๐กโ๐๐๐ ๐ฆ๐๐ข ๐๐๐ ๐๐๐๐๐๐๐ ๐ก๐ ๐๐ ๐๐ฅ๐๐๐๐๐๐ก ๐ก๐ ๐๐๐ก ๐๐๐๐ ๐๐๐ ๐๐ ๐ก๐ ๐ข๐ ๐ log ๐๐๐ค๐
C12 โ 8.5 โ Operation log ๐ ๐๐๐ก๐๐
Log Page 9
4 2 ๐๐๐4 ๐๐๐2 ๐๐๐4 ๐ฅ๐๐๐2
๐๐๐4๐๐๐2โฏโฏโฏโฏ ๐ฅ
log 4 ๐ฅ 4 2 2 2 ๐ฅ 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
Deโlog Both sides
log 4 log ๐ฅ 4 ๐ฅ
3 5 ๐๐๐3 ๐๐๐5 ๐๐๐3 ๐ฅ๐๐๐5
๐๐๐3๐๐๐5โฏโฏโฏโฏ ๐ฅ
log 3 ๐ฅ ๐ฅ 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 deโlog both sides if one log equals one log.
๐ด๐๐๐๐๐๐๐๐ ๐๐๐ ๐ค๐๐
4 7 ๐๐๐4 ๐๐๐7 ๐๐๐4 ๐ฅ 1 ๐๐๐7 ๐๐๐4 ๐ฅ๐๐๐7 log 7๐๐๐4 ๐๐๐7 ๐ฅ๐๐๐7๐๐๐4 ๐๐๐7
๐๐๐7โฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏ ๐ฅ
๐ฅ๐๐๐4 ๐๐๐7
๐๐๐7โฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏ
๐ฅ 0.29
Check Answer:
5 . 3
2 9 ๐๐๐2 ๐๐๐9 2๐ฅ 5 ๐๐๐2 ๐ฅ 2 ๐๐๐9 2๐ฅ๐๐๐2 5๐๐๐2 ๐ฅ๐๐๐9 2๐๐๐9 2๐ฅ๐๐๐2 ๐ฅ๐๐๐9 2๐๐๐9 5๐๐๐2๐ฅ 2๐๐๐2 ๐๐๐9 2๐๐๐9 5๐๐๐2
๐ฅ2๐๐๐9 5๐๐๐22๐๐๐2 ๐๐๐9
โฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏ
DistributeCombine x's on one sideEverything else on other sideFactor out xDivide
๐บ๐ถ๐น ๐ฅDivide
Before you log both sides!
8 3 ๐๐๐8 ๐๐๐3 ๐๐๐8 2๐ฅ๐๐๐3
๐๐๐8๐๐๐3โฏโฏโฏโฏ 2๐ฅ
log 8
2โฏโฏโฏโฏโฏ ๐ฅ
12โฏโฏlog 8 ๐ฅ
x log 8โฏโฏ
Bring Fraction In Front.Bring Coefficient Up
Or
8 2 2๐๐๐8 log 2 2๐๐๐8 ๐๐๐2 ๐๐๐2
Or divide log7 and minus 1
6 3 14 ๐๐๐6 3 ๐๐๐14 ๐๐๐6 ๐๐๐3 ๐๐๐14 ๐๐๐6 ๐ฅ๐๐๐3 2๐ฅ 5 ๐๐๐14 ๐๐๐6 ๐ฅ๐๐๐3 2๐ฅ๐๐๐14 5๐๐๐142๐ฅ๐๐๐14 ๐ฅ๐๐๐3 ๐๐๐6 5๐๐๐14๐ฅ 2๐๐๐14 ๐๐๐3 ๐๐๐6 5๐๐๐14
๐ฅ๐๐๐6 ๐๐๐15
2๐๐๐14 ๐๐๐3โฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏ
๐ ๐ข๐๐ 6 ๐๐๐๐๐ ๐ ๐ฅ log a ๐๐๐๐ฅlog ๐ฅ log ๐ ๐๐๐๐ฅlog ๐ฅ log ๐
๐๐๐๐โฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏ
๐๐๐๐ฅlog ๐โฏโฏโฏโฏ
log ๐ฅ๐๐๐๐ฅ๐๐๐๐โฏโฏโฏโฏ
log ๐ฅ log ๐ฅ ๐ฅ ๐ฅ
C12 โ 8.6 โ Log/Delog Both Sides Notes
Log Page 10
๐ฆ log ๐ฅ
๐ฆ 2
๐ฅ ๐ฆ
๐ฅ ๐ฆ
1 12โฏโฏ
0 1
1 2
๐ฅ ๐ฆ
12โฏโฏ
1
1 0
2 1
๐บ๐๐๐โ:
๐ฆ log ๐ฅ
๐ฆ 2
๐ฆ log ๐ฅ 2 1 ๐บ๐๐๐โ:
๐ฆ 2
๐ฆ 2
๐ฅ ๐ฆ
๐ฆ log ๐ฅ
๐ฆ log ๐ฅ 2 1
๐ฟ๐๐๐ก 2๐๐ 1
๐ท๐๐๐๐๐: ๐ฅ 0
๐ฅ 2 0 ๐ฅ 2
๐๐ด: ๐ฅ 0
๐ฅ 2 0 ๐ฅ 2
C12 โ 8.7 โ Graph Log Notes
Log Page 11
๐ฆ 2
1,12โฏโฏ
0,11,2
2,4
12โฏโฏ, 1
1,02,1
4,2
y log ๐ฅ
Remember: Inverse: Switch x and y Remember: A diagonal reflection over the line ๐ฆ ๐ฅ
๐ฆ 2 ๐ฅ 2 ๐๐๐๐ฅ ๐๐๐2 ๐๐๐๐ฅ ๐ฆ๐๐๐2
๐๐๐๐ฅ๐๐๐2โฏโฏโฏโฏ ๐ฆ
log ๐ฅ ๐ฆ ๐ฆ log ๐ฅ๐ ๐ฅ log ๐ฅ
y log ๐ฅ ๐ฅ log ๐ฆ 2 ๐ฆ ๐ฆ 2๐ ๐ฅ 2
๐ฆ 2 ๐ฅ 2 ๐ฆ log ๐ฅ๐ ๐ฅ log ๐ฅ
๐ฅ ๐ฆ
0 ๐ข๐๐
12โฏโฏ
1
1 0
2 1
2 4
๐ฅ ๐ฆ
1 12โฏโฏ
0 1
1 2
2 4
๐๐ค๐๐ก๐โ ๐ฅ ๐๐๐ ๐ฆLog Both SidesBring Exponents Down In FrontDivide
Change of baseMirrorInverse Function notation
๐๐ค๐๐ก๐โ ๐ฅ ๐๐๐ ๐ฆExponential to ๐๐๐ Form
๐ต๐๐๐ ๐กโ๐ ๐๐กโ๐๐ ๐๐๐ฆ!
๐ฆ 2 3 ๐ฅ 2 3 ๐ฅ 3 2 log ๐ฅ 3 ๐ฆ 1 ๐๐๐2
log ๐ฅ 3
๐๐๐2โฏโฏโฏโฏโฏโฏโฏโฏโฏ ๐ฆ 1
log ๐ฅ 3 ๐ฆ 1log ๐ฅ 3 1 ๐ฆ ๐ฆ log ๐ฅ 3 1 ๐ ๐ฅ log ๐ฅ 3
๐ฆ log ๐ฅ 3 1 ๐ฅ log ๐ฆ 3 1 ๐ฅ 1 log ๐ฆ 3 2 ๐ฆ 32 3 ๐ฆ ๐ฆ 2 3 ๐ ๐ฅ 2 3
๐ผ๐๐ฃ๐๐๐ ๐ ๐๐๐๐๐
C12 โ 8.7 โ Inverse Log Graphs Notes
Log Page 12