11.3 & 11.6 Notes

The natural number e and solving base e exponential equations

11.3 & 11.6 Notes

In this unit of study, you will learn several methods for solving several types of exponential equations.

In previous lessons, you learned how to solve exponential equations using properties of exponents and using logarithms.

In this lesson, you will learn how to solve base e exponential equations.

11.3 & 11.6 Notes

Evaluate for:

a. x = 100

b. x = 1000

c. x = 10000

(round answers to the nearest thousandth)

11

x

x

2.7052.7172.718

11.3 & 11.6 Notes

This is called the natural number e or Euler’s number. It is named for the Swiss mathematician Leonhard Euler for his work in the area of logarithms.

1lim 1

x

x x

e 2.72

11.3 & 11.6 NotesSimilar to π, e is an irrational number. That is, it cannot be expressed as the ratio of integers. Its value is a non-repeating, non-terminating decimal. Similar to π, e is a naturally-occurring mathematical phenomenon that cannot be completely explained.

We know that the ratio of every circle’s circumference to its diameter is approximately 3.14.

Similarly, some exponential growths and decays occur to a base of approximately 2.72.

11.3 & 11.6 Notes

Just as there are exponential equations with integers and rational numbers as bases, there are exponential equations with irrational numbers such as the natural number e as bases.

How are base e exponential equations solved?

Using logarithms.

11.3 & 11.6 Notes

Since e is an irrational number, what base logarithm will be used to solve base e exponential equations?

Do you see a log base e button on your calculator?

Logarithms with the natural number e as their base are called natural logarithms. Natural logarithm is abbreviated ln.

loge

ln loge

11.3 & 11.6 Notes

ln ln 5xe

5xe

ln ln 5x e

1.609x ln logee e

ln loge

1

11.3 & 11.6 Notes – Example 1

ln ln 6xe

6xe

ln ln 6x e

1.792x

11.3 & 11.6 Notes – Practice 1

ln ln 7xe

7xe

ln ln 7x e

1.946x

11.3 & 11.6 Notes – Example 2

41525ln ln

6xe

41525 6 xe

5.538 4 lnx e

1.385x

41525

6xe

11.3 & 11.6 Notes – Practice 2

31249ln ln

175xe

31249 175 xe

1.965 3 lnx e

0.655x

31249

175xe

11.3 & 11.6 Notes – Example 3

2ln ln1xe

214 6 20xe

2 ln 0x e

0x

26 6xe 2 1xe

11.3 & 11.6 Notes – Practice 3

3ln ln 6xe

34 2 8xe

3 ln 1.792x e

0.597x

32 12xe 3 6xe

11.3 & 11.6 Notes – Example 4

2 5x xe e

2 7 10 0x xe e

ln ln 2 ln ln 5x xe e

ln ln 2 ln ln 5x e x e

0

2 0 5 0x xe e

0.693, 1.609x

2 5x xe e

11.3 & 11.6 Notes – Practice 4

2 4x xe e

2 6 8 0x xe e

ln ln 2 ln ln 4x xe e

ln ln 2 ln ln 4x e x e

0

2 0 4 0x xe e

0.693, 1.386x

2 4x xe e