THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.
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Transcript of THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.
![Page 1: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/1.jpg)
THE LAWS OF LOGARITHMS
Patterns & Relations #3
![Page 2: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/2.jpg)
Prerequisites
1. Simplify.
a) x 4 x 2 b) 105 103 c) am an
d) x 3 4 e) 105 2
f) am n
g) x 8
x 4 h) 107
103 i) am
an
2. Evaluate.
a) 271
3 b) 641
2
3. Write a1
n without exponents.
![Page 3: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/3.jpg)
Answers to Prerequisites
1. a) x6 b) 108 c) am+n
d) x12 e) 1010 f) amn
g) x4h) 104 i) am-n
2. a) 3 b) 8
3.
an
![Page 4: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/4.jpg)
Law of Logarithms for Powers
Read p.79 of your text One example is in base 10 The second example is in base a
The law states:
If x and n are real numbers, and x 0, then
loga xn n loga x a 0, a1
![Page 5: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/5.jpg)
Example
3x 20
log3x log20
x log3 log20
x log20
log3
x 2.72683
![Page 6: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/6.jpg)
You try…
Solve the equation 5x = 40 to 5 decimal places.
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The solution
5x 40
log5x log 40
x log5 log 40
x log40
log5
x 2.29203
![Page 8: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/8.jpg)
Another Example
let 9 5x
log9 log5x
log9 x log5
log9
log5x
x 1.36521
![Page 9: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/9.jpg)
You try…
![Page 10: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/10.jpg)
The solution
let 3x 14
log3x log14
x log3 log14
x log14
log3
x 2.40217
![Page 11: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/11.jpg)
Law of Logarithms for Multiplication
Read p.81 of your text One example is in base 10 The second example is in base a
The law states:
If x and y are positive real numbers, then
loga xy loga x loga y a 0, a1
![Page 12: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/12.jpg)
Example
120 1.12n 4000
log(120 1.12n ) log 4000
log120 log1.12n log4000
log120 n log1.12 log4000
n log1.12 log 4000 log120
n log4000 log120
log1.12
n 30.9
![Page 13: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/13.jpg)
You try…
Solve the equation 15 x 1.08n = 20 000 to 1 decimal place.
![Page 14: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/14.jpg)
The solution
15 1.08n 20000
log(15 1.08n ) log20000
log15 log1.08n log20000
log15 n log1.08 log20000
n log1.08 log20000 log15
n log20000 log15
log1.08
n 93.5
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Law of Logarithms for Division
The law states:
If x and y are positive real numbers, then
loga
x
y
loga x loga y a 0, a1
![Page 16: THE LAWS OF LOGARITHMS Patterns & Relations #3. Prerequisites.](https://reader035.fdocuments.us/reader035/viewer/2022062222/56649e7c5503460f94b7dea5/html5/thumbnails/16.jpg)
Example
log64 log640
log64
640
log1
10
log10 1
1
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You try…
Simplify: log550 – log5 0.4
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You try…
log5 50 log5 0.4
log5
50
0.4
log5(125)
3
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Textwork
p.83/ 1-19