The natural number e and solving base e exponential equations 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)

11.3 & 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.

11.3 & 11.6 Notes Similar 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.

11.3 & 11.6 Notes

11.3 & 11.6 Notes – Example 1

11.3 & 11.6 Notes – Practice 1

11.3 & 11.6 Notes – Example 2

11.3 & 11.6 Notes – Practice 2

11.3 & 11.6 Notes – Example 3

11.3 & 11.6 Notes – Practice 3

11.3 & 11.6 Notes – Example 4

11.3 & 11.6 Notes – Practice 4