# Natural Log: Rules & Properties Video

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• 0:00 The Letter E
• 1:07 What Is the Natrual Log?
• 1:11 Rules of Natural Logs
• 2:26 Natural Log Properties
• 2:42 Examples
• 5:05 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
The natural log (ln) is the logarithm to the base 'e' and has extensive uses in science and finance. This lesson will define the natural log as well as give its rules and properties.

## The Letter 'E'

In mathematics, the letter 'e' is used to denote an approximation coming close to 2.72. Just like pi is an approximation coming close to 3.14 and is useful for determining information about a circle, e is used in equations that deal with growth and decay equations. This growth or decay can be with just about anything; bacteria, nuclear waste, or finances are a few examples.

The equation that is used for growth and decay problems is A = Pe^(rt), where 'A', is the ending amount, 'P' is the beginning amount, 'r' is the growth or decay rate (expressed as a decimal), and 't' is the time (in whatever unit was used on the growth/decay rate).

The number 'e' is the natural exponential, because it arises naturally in math and the physical sciences, just as pi arises naturally in geometry. This number was discovered in the 1700's by Leonhard Euler, a Swiss mathematician.

## What is The Natural Log

The natural log (ln) is the inverse operation of e, the natural exponent.

## Rules of Natural Logs

There are rules that govern the way natural logs work. They are similar to the rules for other logarithms.

The ln of the multiplication of x and y is the sum of the ln of x and ln of y.

ln(x)( y) = ln(x) + ln(y)

For example:

ln(3)( 7) = ln(3) + ln(7)

Quotient rule

The ln of the division of x and y is the difference of the ln of x and ln of y.

ln(x / y) = ln(x) - ln(y)

For example:

ln(3 / 7) = ln(3) - ln(7)

Power rule

The ln of x raised to the power of y is y times the ln of x.

ln(x^y) = y x ln(x)

For example:

ln(2^8) = 8 x ln(2)

## Natural Log Properties

There are also properties that apply to natural logarithms.

The ln(0) is undefined

The ln(1) = 0

The ln(e) = 1

## Examples

1.) Simplify ln(2)

To solve this problem, simply use your calculator. There should be a button for 'ln' that will give you the information you need.

In this case, the answer is:

ln(2) = 0.693

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