Week 3 *: Logaritmen *
The natural logarithm
Euler's number, denoted by #\e#, was introduced in the theory of sequences as an infinite sum. It is a real number with approximation #\e\approx 2.71828182846#. We will show here why it is such a special number.
By #\exp# we denote the exponential function, which is given by #\exp(x) = {\e}^x#.
By #\ln# we denote the inverse function of #\exp#.
The function #\exp# is called the natural exponential function and #\ln# the natural logarithm.
These two functions have special names because of the following special properties.
The derivative of the function #\exp(x)# is #\exp(x)#.
The derivative of the function #\ln(x)# is #\dfrac{1}{x}#.
First we determine the derivative of #\exp# in #c#. The difference quotient can be rewritten as follows. \[\dfrac{\exp(x)-\exp(c)}{x-c} = \dfrac{{\e}^x- {\e}^c}{x-c} = {\e}^c\dfrac{{\e}^{x-c}-1}{x-c}.\]
From Power series we know that #\lim_{h\to 0}\dfrac{{\e}^h-1}{h} = 1#. From this follows \[\exp'(c) = \lim_{x\to c}\dfrac{\exp(x)-\exp(c)}{x-c} = \lim_{x\to c}{\e}^c\dfrac{{\e}^{x-c}-1}{x-c} = {\rm e}^c\lim_{x\to c}\dfrac{{\e}^{x-c}-1}{x-c} = {\e}^c=\exp(c)\tiny.\] Hence, the derivative of #\exp# is #\exp# itself.
Next we define the derivative of the function #\ln(x)#. We use the definition of #\ln# as #\exp^{-1}#, the inverse function of #\exp#.
\[\left(\ln(x)\right)' = \left(\exp^{-1}(x)\right)'=\dfrac{1}{\exp'\left(\exp^{-1}(x)\right)}=\dfrac{1}{\exp\left(\exp^{-1}(x)\right)}=\dfrac{1}{x}\tiny\tiny.\]In the first step we use the rule for differentiating inverse functions, in the second step we use #\exp'=\exp#, and in the last step we use the definition of inverse functions.
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