Exponential Functions and Natural Logarithmic Functions
We have learned a lot of
10 based logarithms. The logarithm of the form is called the general logarithm.
In addition to the 10-based logarithm, there is a logarithm based on the number
e (e = 2, ...) called the natrural logarithm. The number e is an irrational
number determined by Leunard Euler (1707 - 1783) and published in 1728.
The number e is defined
as:
Exponential Limit Approach Table
The e
value can be obtained by the steps in the following 
approach table,
approach table,
x
|
1 + x
|
lim(1+ x)^1/x
|
0.1
|
1.1
|
2.59374246
|
0.01
|
1.01
|
2.70481381
|
0.001
|
1.001
|
2.71692384
|
0.0001
|
1.0001
|
2.71814591
|
0.00001
|
1.00001
|
2.71825465
|
0.000001
|
1.000001
|
2.71828182
|
-0.1
|
0.9
|
2.86797198
|
-0.01
|
0.99
|
2.73199901
|
-0.001
|
0.999
|
2.71964213
|
-0.0001
|
0.9999
|
2.71841775
|
-0.00001
|
0.99999
|
2.71828182
|
-0.000001
|
0.999999
|
2.71828182
|
It appears that the number e approaches the value 2.71828182
You have learned various
forms of function. With the number "e" we can formulate another function,
namely f : x → ex, x ∈ R, which is called an exponential function. The function
is very much helpful in various kinds of problems such as growth problems or continuous
depreciation at any time (Continuous depreciation).
Graph f(x) = ex
The value of e is between
2 and 3, so the function graph f(x) = ex can be shown with the curve
between y = 2x and y = 3x,
As in the function graph f(x)
= ax, a > 0, a ≠ 1, the following properties are obtained,
Graph through points (0,
1).
The graph is always above
the X-axis.
Natural Logarithmic Function
Natural logarithms are
logarithms based on the number “e”, written with the following notation,
elog e is
expressed as ln x
or ln x = elog x
“be read len x”
Example 1.
Ln 1 = 0 (because e0 = 1)
Ln e = 1 (because e1 = e)
Ln 1/e = -1 (because e-1 = 1/e)
Ln e2 = 2 (one
of the properties of logarithms)
Example 2.
Find the equation from 5ex-3 = 4
Answer:
5ex-3 = 4
ex-3 = 4
ln ex-3 = 0.8
(x - 3) ln e = ln 0.8
x – 3 = ln 0.8
x = 3 + ln 0.8
The function f: x → ln x, x > 0 is called the natural logarithmic function.
As with other number-based
logarithmic functions, the function graph f(x) = ln x can be presented in the
figure below,
The graph y = ln x can be
obtained from the reflection of the graph y = ex with respect to the
line y = x.,
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