Download - Notes 6-3
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Section 6-3L o g a r i t h m s
![Page 2: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/2.jpg)
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
![Page 3: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/3.jpg)
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −4
![Page 4: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/4.jpg)
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = −4
![Page 5: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/5.jpg)
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = 0 a = −4
![Page 6: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/6.jpg)
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = 0
a = 1
2
a = −4
![Page 7: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/7.jpg)
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = 0
a = 1
2 a = 2
a = −4
![Page 8: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/8.jpg)
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = 0
a = 1
2 a = 2
a = 11
a = −4
![Page 9: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/9.jpg)
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = 0
a = 1
2 a = 2
a = 11 No solution
a = −4
![Page 10: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/10.jpg)
Definition of Logarithm
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Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
![Page 12: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/12.jpg)
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
y = log
bx IFF by = x
![Page 13: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/13.jpg)
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
What does this mean?
y = log
bx IFF by = x
![Page 14: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/14.jpg)
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
What does this mean?
y = log
bx IFF by = x
y = log
bx IFF by = x
![Page 15: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/15.jpg)
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
What does this mean?
y = log
bx IFF by = x
Base
y = log
bx IFF by = x
![Page 16: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/16.jpg)
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
What does this mean?
y = log
bx IFF by = x
Base
Exponent
y = log
bx IFF by = x
![Page 17: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/17.jpg)
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
![Page 18: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/18.jpg)
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
−1
![Page 19: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/19.jpg)
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
![Page 20: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/20.jpg)
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
![Page 21: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/21.jpg)
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
![Page 22: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/22.jpg)
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Why?
![Page 23: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/23.jpg)
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Why?
62 = 36
![Page 24: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/24.jpg)
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Why?
62 = 36
25
![Page 25: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/25.jpg)
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Why?
62 = 36
25
Why?
![Page 26: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/26.jpg)
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Why?
62 = 36
25
Why?
625 = 365 = 625
![Page 27: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/27.jpg)
Example 2Evaluate.
log
9243
![Page 28: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/28.jpg)
Example 2Evaluate.
log
9243
92 = 81
![Page 29: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/29.jpg)
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729
![Page 30: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/30.jpg)
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
![Page 31: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/31.jpg)
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
![Page 32: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/32.jpg)
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
2435 = 3
![Page 33: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/33.jpg)
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
2435 = 3 = 912
![Page 34: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/34.jpg)
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
Ok, what does that mean? 2435 = 3 = 9
12
![Page 35: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/35.jpg)
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
Ok, what does that mean? 2435 = 3 = 9
12
9
12( )5
= 243
![Page 36: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/36.jpg)
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
Ok, what does that mean? 2435 = 3 = 9
12
9
12( )5
= 243
log
9243 = 5
2
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Common Logarithms
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Common Logarithms
Logarithms with a base of 10
![Page 39: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/39.jpg)
Common Logarithms
Logarithms with a base of 10
You will see this one on your calculator
![Page 40: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/40.jpg)
Example 3Solve to the nearest hundredth.
10y = 73
![Page 41: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/41.jpg)
Example 3Solve to the nearest hundredth.
10y = 73
Ok, let’s rewrite this as a logarithm.
![Page 42: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/42.jpg)
Example 3Solve to the nearest hundredth.
10y = 73
Ok, let’s rewrite this as a logarithm.
log 73 = y
![Page 43: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/43.jpg)
Example 3Solve to the nearest hundredth.
10y = 73
Ok, let’s rewrite this as a logarithm.
log 73 = y
![Page 44: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/44.jpg)
Example 3Solve to the nearest hundredth.
10y = 73
Ok, let’s rewrite this as a logarithm.
log 73 = y
![Page 45: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/45.jpg)
Example 3Solve to the nearest hundredth.
10y = 73
Ok, let’s rewrite this as a logarithm.
log 73 = y
y ≈ 1.86
![Page 46: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/46.jpg)
Example 4Solve log t = 2.9 to the nearest tenth.
![Page 47: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/47.jpg)
Example 4Solve log t = 2.9 to the nearest tenth.
Rewrite as a power.
![Page 48: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/48.jpg)
Example 4Solve log t = 2.9 to the nearest tenth.
Rewrite as a power.
102.9 = t
![Page 49: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/49.jpg)
Example 4Solve log t = 2.9 to the nearest tenth.
Rewrite as a power.
102.9 = t
t ≈ 794.3
![Page 50: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/50.jpg)
Properties of Logarithms
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Properties of Logarithms
Domain is the set of positive real numbers.
![Page 52: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/52.jpg)
Properties of Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
![Page 53: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/53.jpg)
Properties of Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
(1, 0) will be on the graph; logb1 = 0.
![Page 54: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/54.jpg)
Properties of Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
(1, 0) will be on the graph; logb1 = 0.
The function is strictly increasing.
![Page 55: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/55.jpg)
Properties of Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
(1, 0) will be on the graph; logb1 = 0.
The function is strictly increasing.
As x increases, y has no bound.
![Page 56: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/56.jpg)
Properties of Logarithms
![Page 57: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/57.jpg)
Properties of Logarithms
As x gets smaller and approaches 0, the values of the function are negative with larger absolute values.
That means when x is between 0 and 1, the exponent will be negative.
![Page 58: Notes 6-3](https://reader031.vdocuments.mx/reader031/viewer/2022013011/556674cdd8b42a0f168b4595/html5/thumbnails/58.jpg)
Properties of Logarithms
As x gets smaller and approaches 0, the values of the function are negative with larger absolute values.
That means when x is between 0 and 1, the exponent will be negative.
The y-axis is an asymptote.
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Homework
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Homework
p. 387 #1 - 26