Exponential functions and logarithms: The base e and the natural logarithm
The base e and the natural logarithm
The number #\euler\approx 2.71828182846\ldots#, also called Euler's number, is an important number in mathematics that has quite a few special properties, as you will see later in the differentiation chapter. There are several definitions for #\e#.
#\begin{array}{rclcl}
1+\frac{1}{1} + \frac{1}{1 \cdot 2} + \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \ldots & = & \orange{\e} &\phantom{x}& \blue{\text{Sum of infinite terms (i.e. an infinite series)}} \\ \\
\lim\limits_{n \to \infty} \big( 1+ \frac{1}{n} \big)^n & = & \orange{\e} & & \blue{\text{Used in the study of compound interest}} \\ \\
\lim\limits_{h \to 0} \dfrac{\orange{\e}^h - 1}{h} & = & 1 && \orange{\e}\blue{\text{ is the unique number for which this limit equals }1}
\end{array}#
The number #\e# can be used as the base in an exponential expression, #\e^n#, then we use the rules for exponents.
We can take the number #\e# as the base of a logarithm. This logarithm is called the natural logarithm and it is denoted by #\ln#.
The natural logarithm
The natural logarithm is
\[\ln(\blue{x})=\log_\orange{\e}(\blue{x})\]
Example
\[\begin{array}{rcl}\\ \ln(\orange{\e}^\blue{x})&=&\blue{x}\\ \end{array}\]
With the natural logarithm we use the same rules as with other logarithms. Rewriting to different base numbers goes the same as with the natural logarithm.
Rewrite the following expression, simplifying as far as possible.
\[\log_4(\e)\cdot\ln(4)\]
#\begin{array}{rcl}\log_4(\e)\cdot\ln(4)&=&\dfrac{\ln(\e)}{\ln(4)}\cdot\ln(4)\\
&&\phantom{xxx}\blue{\log_a\left(x\right)=\dfrac{\log_b\left(x\right)}{\log_b\left(a\right)}}\\
&=&\ln\left(\e\right)\\
&&\phantom{xxx}\blue{\text{simplified}}\\
&=&1\\
&&\phantom{xxx}\blue{\ln(\e)=1}
\end{array}#
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