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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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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.

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 = P*e*^(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.

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

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)

There are also properties that apply to natural logarithms.

The ln(0) is undefined

The ln(1) = 0

The ln(*e*) = 1

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

2.) **Simplify** ln(3^3).

We can use the power rule to solve this problem.

(3)ln(3) = (3)(1.099)

3.296

Another way to solve this problem would be to calculate 3^3 first. Then solve

ln(27) = 3.296

3.) **A certain strain of bacteria grows from 50 to 450 in 120 minutes. Find 'k' for the growth formula y = a e^(kt) where t is in minutes**.

To solve this problem, first substitute the information that you know into the equation

y = 450

a = 50

t = 120 minutes

450 = 50*e*^(120k)

Next, divide both sides by 50 so that * 'e' * is alone on one side of the equation.

9 = *e*^(120k)

Then, take the ln of both sides of the equation.

ln(9) = ln(*e*^(120k))

The ln(9) can be solved using your calculator.

2.197 = ln(*e*^(120k))

The exponent is moved out front on the right side to get:

2.197 = 120k ln(*e*)

The ln(*e*) = 1, so the equation becomes very easy to solve.

2.197 = 120k

k = 0.018

'k' is a constant that can then be used to determine other things about this strain of bacteria, such as how many there will there be after 48 hours, or how long will it take the population to reach 5,000?

Let's review! The **natural log** (ln) is the inverse of the natural exponent, which is represented by the letter * 'e' *. It is an approximation that is used in growth and decay problems in both science and finance.

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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

- What is a Function: Basics and Key Terms 7:57
- Graphing Basic Functions 8:01
- Compounding Functions and Graphing Functions of Functions 7:47
- Understanding and Graphing the Inverse Function 7:31
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