the natural logarithm
natural logarithmic function
inverse function
Given the graph of the e-function $${y = e^x}$$, it is possible to rearrange the equation and make x the subject. The result is the logarithm to the base e.
$${\log_e y = x}$$
Which is now know as the natural logarithm and written as:
$${\ln y = x }$$
Swapping x and y which geometrically means reflecting the exponential function on the line y = x we then have the natural logarithmic function, the inverse function of the e-function.
$${f(x) = \ln x}$$
The domain (Definitionsbereich)
$${ \mathbb{D}_{f} = {\{x \in \mathbb{R} , | ,x > 0 \} }}$$
and range (Wertebereich)
$${ \mathbb{W}_{f} = {\{x \in \mathbb{R} \} }}$$
make this possible for both functions.
| $${f(x)}$$ | Definitionsbereich (domain) | Wertebereich (range) |
|---|---|---|
| $${f(x) = \ln x}$$ | $${{\{x \in \mathbb{R} , | ,x > 0\}}}$$ | $${x \in \mathbb{R} }$$ |
| $${f(x) = e^x}$$ | $${x \in \mathbb{R} }$$ | $${{\{x \in \mathbb{R} , | ,x > 0\}}}$$ |
the derivative - Die Ableitung
To find the derivative, we will use a method called implicit differentiation.
Let
$${y = \ln x}$$
then $$e^y = x$$
Now, the implicit differentiation, means we derive both sides with respect to x.
So,
$$e^y \cdot y^{\prime} = 1$$ (applying the chain rule on the left hand side)
Rearrange the equation to make y’ the subject and substitute $e^y = x$
Hence
$$ y^{\prime} = \frac{1}{x}$$
or
$$ [\ln{x}]^{\prime} = \frac{1}{x}$$
Historically, there is a connection between the area under the graph of $ y = \frac{1}{x}$ and the natural logarithm.
The area from x=1 to x=e under the graph of $ y = \frac{1}{x}$ is equal to 1.