8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.
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Transcript of 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.
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8.5 Natural Logarithms
©2001 by R. Villar
All Rights Reserved
![Page 2: 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.](https://reader036.fdocuments.in/reader036/viewer/2022082817/56649de85503460f94ae1f66/html5/thumbnails/2.jpg)
Natural Logarithms
Natural Logarithm: if x is a positive real number, then the natural logarithm of x is denoted by
loge x or ln x
A function given by f(x) = a + ln bx is called a natural logarithm function.
Example: Use your calculator to find ln 3
Your scientific calculator has a natural logarithm key on it.
ln 3 = 1.0986
Let’s look at the graph of a natural logarithm function...
![Page 3: 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.](https://reader036.fdocuments.in/reader036/viewer/2022082817/56649de85503460f94ae1f66/html5/thumbnails/3.jpg)
Ex. Graph f(x) = 3 – ln (x – 1)
x
f(x)
![Page 4: 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.](https://reader036.fdocuments.in/reader036/viewer/2022082817/56649de85503460f94ae1f66/html5/thumbnails/4.jpg)
Ex. Graph f(x) = 3 – ln (x – 1)
x
f(x)
1.25
![Page 5: 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.](https://reader036.fdocuments.in/reader036/viewer/2022082817/56649de85503460f94ae1f66/html5/thumbnails/5.jpg)
Ex. Graph f(x) = 3 – ln (x – 1)
x
f(x)
1.25 1.5 2 3 4 5 6
![Page 6: 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.](https://reader036.fdocuments.in/reader036/viewer/2022082817/56649de85503460f94ae1f66/html5/thumbnails/6.jpg)
Ex. Graph f(x) = 3 – ln (x – 1)
x
f(x)
1.25 1.5 2 3 4 5 64.38
![Page 7: 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.](https://reader036.fdocuments.in/reader036/viewer/2022082817/56649de85503460f94ae1f66/html5/thumbnails/7.jpg)
Ex. Graph f(x) = 3 – ln (x – 1)
x
f(x)
1.25 1.5 2 3 4 5 64.38 3.69 3 2.31 1.90 1.611.39
![Page 8: 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.](https://reader036.fdocuments.in/reader036/viewer/2022082817/56649de85503460f94ae1f66/html5/thumbnails/8.jpg)
Ex. Graph f(x) = 3 – ln (x – 1)
x
f(x)
1.25 1.5 2 3 4 5 64.38 3.69 3 2.31 1.90 1.611.39
![Page 9: 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.](https://reader036.fdocuments.in/reader036/viewer/2022082817/56649de85503460f94ae1f66/html5/thumbnails/9.jpg)
Ex. Graph f(x) = 3 – ln (x – 1)
x
f(x)
1.25 1.5 2 3 4 5 64.38 3.69 3 2.31 1.90 1.611.39
The line x = 1 is the vertical asymptote of The function.
![Page 10: 8.5 Natural Logarithms ©2001 by R. Villar All Rights Reserved.](https://reader036.fdocuments.in/reader036/viewer/2022082817/56649de85503460f94ae1f66/html5/thumbnails/10.jpg)
Condense the expressions:a. ln 18 – ln 3
b. 3ln x + ln y
c.
= ln 6
= ln x3y
= ln 41/2 + 2 ln 3
= ln 2 + ln 32
= ln 2 + ln 9
= ln 18
12ln4+2(ln6−ln2)
Natural logarithms can be condensed/expanded using the properties of logarithms: