Natural Logarithms

After $\pi$, the next weird number is called $e$, for $\textit{exponential}$. It was first discussed by Jacob Bernoulli in 1683. It occurs in problems about compound interest, leds to logarithms, and tells us how variables like radioactivity, temperature, or the human population increase or decrease.

In 1614 John Napier knew, from personal experience, that many scientific problems, especially in astronomy, required multiplying complicated numbers together, or finding square roots and cube roots. As a time when there was no electricity, let alone computers, calculations had to be done by hand. Adding two decimal numbers together was reasonably simple, but multiplying them was much harder. So Napier invented a method for turning multiplication into addition. The trick was to work with the powers of a fixed number.

Since many exponential models have base $e$, it is useful to also consider the logarithm in base $e$, called the $\textit{natural logarithm}$.

The $\textit{natural logarithm}$ of a positive number is its power of $e$.
The natural logarithm of $x$ is written $\log_{e}{x}$ or $\ln{x}$.

For example, $e^{2.1} \approx8.166 \cdots$, so $\log_{e}{8.166} \approx 2.1$, or $\ln{8.166} \approx 2.1$.
$$e^x = y \Leftrightarrow x = \log_{e}{y}$$

Example 1

Write an equivalent statement for $e^2 \approx 7.389$.

$\log_{e}{7.389} \approx 2$

Example 2

Write an equivalent statement for $\log_{e}{25} \approx 3.22$.

$e^{3.22} \approx 25$

From the rule $e^x = y \Leftrightarrow x = \log_{e}{y}$, we obtain that:
\( \begin{align} \displaystyle
e^x &= y \cdots (1) \\
x &= \ln{y} \cdots (2) \\
\color{green} x &\color{green}= \color{green}\ln{e^x} &\text{substitute } (1) \text{ into } (2) \\
\color{green}e^{\ln{y}} &\color{green}= \color{green}y &\text{substitute } (2) \text{ into } (1) \\
\end{align} \)

Example 3

Simplify $\ln{e^4}$.

$\ln{e^4} = 4$

Example 4

Simplify $e^{\ln{5}}$.

$e^{\ln{5}} = 5$

Example 5

Simplify $\ln{\dfrac{1}{e^3}}$.

\( \begin{align} \displaystyle
\ln{\dfrac{1}{e^3}} &= \ln{e^{-3}} \\
&= -3 \\
\end{align} \)

Example 6

Find $x$, if $10=e^x$.

\( \begin{align} \displaystyle
10 &= e^{\ln{10}} \\
&= e^{2.30 \cdots} \\
\therefore x &= 2.30 \cdots \\
\end{align} \)

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