Download Base e and Natural Logarithms

Base e and Natural Logarithms
Although common logarithms are frequently used for numerical calculations, natural
logarithms are used in most other applications, such as those involving growth and
decay. Natural logarithms are logarithms to base e and is indicated by the
abbreviation ln. The number e is a very important irrational number used extensively
in mathematics. The approximate value of e is 2.718281828.
The natural logarithm of e to any power is that exponent. Thus, powers of e are the
only numbers for which the natural logarithms are integers.
Examples of natural logarithms of e: ln e3 = 3
ln e2 = 2
ln e1 = 1
ln e0 = ln 1 = 0
ln e-1 = -1
ln e-2 = -2
For those natural logarithms other than e, we will use a scientific calculator. For
most calculators, the natural logarithm key is
enter the number first and then press
ln
ln
. For some calculators, you
while for others you press
ln
first and then enter the number – it depends on the brand of your calculator.
Example 1: Use a calculator to approximate ln 55 to four decimal places
ln 55 ≈ 4.0073
Modified from Intermediate Algebra, by Andrew Gloag, Anne Gloag, and Mara Landers, CK-12
Foundation, CC-BY 2013. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by/3.0)
Example 2: Use a calculator to approximate ln 0.4 to four decimal places
ln 0.4 ≈ -0.9163
To find a power of e, we need to use a scientific calculator. On most calculators, e
is a second function calculation whereby we have to use the 2nd or shift key and then
the ln key.
Example 3: Use a calculator to approximate e2.45 to four decimal places
e2.45 ≈ 11.5883
Example 4: Use a calculator to approximate e -1.5 to four decimal places
e -1.5 ≈ 0.2231
To get a clearer understanding of base e functions and natural logarithmic functions,
it is important to graph these functions.
Example 5: Graph f (x) = y = e x
Select values for x and solve for y (use decimal approximations)
x=2
x=1
x=0
x = -1
x = -2
e2 ≈ 7.4
e1 ≈ 2.7
e0 = 1
e -1 ≈ 0.4
e -2 ≈ 0.1
y ≈ 7.4
y ≈ 2.7
y=1
y ≈ 0.4
y ≈ 0.1
Modified from Intermediate Algebra, by Andrew Gloag, Anne Gloag, and Mara Landers, CK-12
Foundation, CC-BY 2013. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by/3.0)
Construct a table of values and graph:
x
2
1
0
-1
-2
y
7.4
2.7
1
0.4
0.1
Example 6: Graph g (x) = y = e 2x
Select values for x and solve for x (use decimal approximations)
x=2
x=1
x=0
x = -1
x = -2
e 2(2) = e 4 ≈ 54.6
e 2(1) = e 2 ≈ 7.4
e 2(0) = e 0 = 1
e 2(-1) = e -2 ≈ 0.1
e 2(-2) = e -4 ≈ 0.02
y ≈ 54.6
y ≈ 7.4
y=1
y ≈ 0.1
y ≈ 0.02
Construct a table of values and graph:
x
2
1
0
-1
-2
y
54.6
7.4
1
0.1
0.02
Modified from Intermediate Algebra, by Andrew Gloag, Anne Gloag, and Mara Landers, CK-12
Foundation, CC-BY 2013. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by/3.0)
Example 8: Graph f (x) = y = ln x
Select values for x and solve for y (use decimal approximations)
x=4
x=2
x=1
x = 0.5
ln 4 ≈ 1.4
ln 2 ≈ 0.7
ln 1 = 0
ln 0.5 ≈ -0.7
y ≈ 1.4
y ≈ 0.7
y=0
y ≈ -0.7
Construct a table of values and graph:
x
4
2
1
0.5
y
1.4
0.7
0
-0.7
Example 9: Graph f (x) = y = ln (x+2)
Select values for x and solve for y (use decimal approximations)
x=2
x=0
x = -1
x = -1.5
ln (2+2) = ln 4 ≈ 1.4
ln (0+2) = ln 2 ≈ 0.7
ln (-1+2) = ln 1 = 0
ln (-1.5+2) = ln 0.5 ≈ -0.7
y ≈ 1.4
y ≈ 0.7
y=0
y ≈ -0.7
Construct a table of values and graph:
x
4
2
1
0.5
y
1.4
0.7
0
-0.7
Modified from Intermediate Algebra, by Andrew Gloag, Anne Gloag, and Mara Landers, CK-12
Foundation, CC-BY 2013. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by/3.0)