Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an...
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![Page 1: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/1.jpg)
Logarithmic Functions and Their Graphs
![Page 2: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/2.jpg)
Consider xf x a
This is a one-to-one function, therefore it has an inverse.
The inverse is called a logarithm function.
Example:416 2 24 log 16 Two raised to what power
is 16?
The most commonly used bases for logs are 10: 10log logx x
and e: log lne x x
lny x is called the natural log function.
logy x is called the common log function.
![Page 3: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/3.jpg)
Definition of Logarithmic Function
b > 0; b 1
Logarithmic Form Exponential Form
y = logb x x = by
![Page 4: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/4.jpg)
The log to the base “b” of “x” is the exponent to which “b” must
be raised to obtain “x”
y = log10 x
y = log e x
x = 10 y
x = e y
![Page 5: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/5.jpg)
Change from Logarithmic To Exponential Form
Log 2 8 = 3 8 = 23
5 = 25 ½Log 25 5 = ½
![Page 6: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/6.jpg)
Change from Exponential To Form Logarithmic
49 = 7 2 log 7 49 = 2
1/5 = 5 –1 log 5 (1/5) = -1
![Page 7: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/7.jpg)
Properties of Logarithmic FunctionsIf b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: Log15 1 = 0
Log10 10 = 1
Log5 5x = x
3log x = x 3
150 = 1
101 = 10
5x = 5x
![Page 8: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/8.jpg)
The Decibel Scale
The decibel level D of a sound of intensity I , measured in watts per
square meter (W/ m2) is given by
where I0 = 10–12 W/ m2 is the intensity of the least audible sound that
an average healthy person can hear.
Sound Intensity, W/ m2 Sound
1.0 10–12 Threshold of hearing
5.2 10–10 Whisper
3.2 10–6 Normal conversation
8.5 10–4 Heavy traffic
3.2 10–3 Jackhammer
1.0 100 Threshold of pain
8.3 102 Jet plane with afterburner
0
log10DI
I
![Page 9: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/9.jpg)
The magnitude M on the Richter scale of an earthquake that releases energy E , measured in joules, is given by
where E0 = 104.40 joules is the energy released by a small reference
earthquake.
0EE
log32
M
Magnitude on Richter scale Destructive power
M < 4.5 Small4.5 < M < 5.5 Moderate5.5 < M < 6.5 Large6.5 < M < 7.5 Major7.5 < M Greatest
The Richter Scale
![Page 10: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/10.jpg)
Since logs and exponentials are inverses the domain and range switch!…the x values and y values are exchanged…
![Page 11: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/11.jpg)
![Page 12: Logarithmic Functions and Their Graphs. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function.](https://reader034.fdocuments.in/reader034/viewer/2022050909/56649dde5503460f94ad7815/html5/thumbnails/12.jpg)
f
x y = 2 x
–31
8
–21
4
–11
2
0 1
1 2
2 4
3 8
f –1
x = 2 y
1
8 –3 1
4 –2 1
2 –1
1 0
2 1
4 2
8 3
Ordered pairs reversed
y
x
y
5 10 –5
5
10
–5
f -1
x = 2y
or y = log2x
f y = 2x
y = x
DOMAIN of = (– , ) = RANGE of
RANGE of f = (0, ) = DOMAIN of
Logarithmic Function with Base 2
f
f -1
f -1