Intro to Natural Log (Base e)

Laws of Logs $\color{purple}{\text{ft. e}}$

Here's a summary of all the useful algebraic properties of logs that you can use, adjusted for the natural logarithm!

Basic facts and identities:

1. $\ln 1=0$
2. $\ln e=1$
3. $\ln e^b=b$
4. $e^{\ln b}=b$

Operations:

1. $\ln bc=\ln b+\ln c$
2. $\ln\frac{b}{c}=\ln b-\ln c$
3. $\ln b^c=c\ln b$
4. $\ln c=\frac{1}{b}\ln c$

Change of base:

1. $\ln b=\frac{\log_{c}b}{\log_{c}e}$

Many times, when we are dealing with natural logarithms, we look for values that are approximately equal to their actual values. Alternatively, you can use a graphing calculator to find the exact values.

The natural log is the special name for a logarithm with base $e.$

Namely, the natural log of $k$ is the solution to the equation $e^x=k$.

The natural log of $k$ is represented as either $\log_{e}(k)$, or, more commonly, shortened to $\ln(k)$.

Properties

From the definition of logarithms, we can derive some simple properties, such as $$\begin{align*} e^{\ln(x)} = x \end{align*}$$ and $$\begin{align*} \ln(e^{x}) = x \end{align*}$$

More properties

The natural log is called natural since it commonly shows up in mathematics.

For example, differentiating a logarithmic function with base $b$, surprisingly, yields the natural log in its answer. $\frac{d}{dx}\log_{b}(x) = \frac{1}{x \: \ln(b)}$

Another interesting property of natural logarithms relating to differentiation is that $\frac{d}{dx}\ln(x) = \frac{1}{x}$, which can be drawn from the previous property.

Numerical value

There are many known ways of deriving the exact value for $e$.

Some of them include:

$$\begin{align*} \sum_{k=0}^{\infty} \frac{1}{k!} \end{align*}$$

and $$\begin{align*} \lim_{x\to\infty} \Big(1+\frac{1}{n} \Big)^n \end{align*}$$

(Video) Properties of Logarithms

by PatrickJMT

Here's a video by PatrickJMT giving you some properties of logarithms and explaining what the natural log is.

$\large{\text{Summary}}$

He then goes over some problems. In the video he goes over the cancellation laws of logarithms. These are $\log_a(a^x)=x$ and $a^{log_a(x)}=x$. The natural log is when you have $\log_e(x)$ and this can be rewritten as $\ln(x)$.

Common Logs and Natural Logs

The natural log is just a log to the base $e$. The graph of the log function is shown in Figure $1$. The log function approaches infinity as $x$ approaches $0$. And when $x=1$ the log equals $0$.

The graph of the natural log is shown in Figure 2. It is a similar shape as the log function but expanded more. The same properties still hold though. As $x$ approaches 0, the function approaches infinity, only it approaches slower than the log function. As always, when $x=1$ the function equals 0.

Natural Logarithms

Next in the series on logarithms and exponential functions, we will tackle natural logs. This is a common logarithm that you will see frequently. The mathematical constant e arises naturally in continuously compounded interest.

We previously learned about Euler's number, $e$, and about logarithms. We can combine the two concepts to learn about natural logarithms.

We can represent the natural logarithm with "ln". "Log base $e$ of $x$" can be written as:

$$log_e (x) = ln(x)$$

The natural logarithm answers the question: "To what power must $e$ be raised to in order to get a given number?"

The natural logarithm is the inverse of the exponential function $y=e^x$. Let's look at what this inverse relationship looks like on a graph!

Just as subtraction undoes addition, the natural logarithm undoes exponentiation with base $e$. They are inverse functions!

$\ln(e^x) = x$ is the inverse of $e^{\ln(x)} = x$

There are some important restrictions on the domain and range of a natural log function. For the domain, the input must be positive ($x>0$). That means that you cannot take the natural log of zero or a negative number.

For the range, the output can be any real number. That means that the range is $(-\infty, \infty)$.

The graph of the natural logarithm has the following properties:

• Passes through the point $(1,0)$
• Increases slowly as $x$ increases
• Has a vertical asymptote at $x=0$