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,

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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