### 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#, we'll give one here.

Euler's number defined by a sum of infinite terms:

\[

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}

\]

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 way with the natural logarithm.

Rewrite the following expression, simplifying as far as possible.

\[\ln(6)\cdot\log_6(\e)\]

#\begin{array}{rcl}\ln(6)\cdot\log_6(\e)&=&\ln(6)\cdot\dfrac{\ln(\e)}{\ln(6)}\\

&&\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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