Chapter 4 4-6 The Natural base, e

Quiz#1O Express each as a single

logarithm.O 1.

O Simplify.O 3O 4. O 5.

Quiz#1O Solve.O 6. O 7. O 8. lo g7 ( 5x+3 )= 3O 9. log ( 3x + 1 ) - log 4 = 2O 10. lo g4 (x-1)+ lo g4 (3x- 1)=2

Objectives

Use the number e to write and graph exponential functions representing real-world situations.

Solve equations and problems involving e or natural logarithms.

Natural BaseO Recall the compound interest

formula A = P(1 + r/n )nt, where A is the amount, P is the principal, r is the annual interest, n is the number of times the interest is compounded per year and t is the time in years.

Natural baseO Suppose that $1 is invested at 100%

interest (r = 1) compounded n times for one year as represented by the function f(n) = P(1 + 1/n )n.

Natural baseO As n gets very large, interest is

continuously compounded. Examine the graph of f(n)= (1 + 1/n )n. The function has a horizontal asymptote. As n becomes infinitely large, the value of the function approaches approximately 2.7182818…. This number is called e. Like , the constant e is an irrational number.

Natural base

Exponential functionsO Exponential functions with e as a

base have the same properties as the functions you have studied. The graph of f(x) = ex is like other graphs of exponential functions, such as f(x) = 3x

The domain of f(x) = ex is all real numbers. The range is {y|y > 0}.

Example 1 O Graph f(x) = ex–2 + 1.

O Make a table. Because e is irrational, the table values are rounded to the nearest tenth.

Example 2O Graph f(x) = ex – 3.

Student guided practice

O Do problems 2-5 in your book page 278

Natural logsO A logarithm with a base of e is called

a natural logarithm and is abbreviated as “ln” (rather than as log

e). Natural logarithms have the

same properties as log base 10 and logarithms with other bases.

Natural logsO The natural logarithmic function

f(x) = ln x is the inverse of the natural exponential function f(x) = ex.

Natural logsO The domain of f(x) = ln x is {x|x >

0}.O The range of f(x) = ln x is all real

numbersO All of the properties of logarithms

from Lesson 4-4 also apply to natural logarithms.

Example #3O Simplify.O A. ln e0.15t

O B. e3ln(x +1)

O C. ln e2x + ln ex

Student guided practice

O Do problems 6-10 in your book page 278

ContinuouslyCompounded

O The formula for continuously compounded interest is A = Pert, where A is the total amount, P is the principal, r is the annual interest rate, and t is the time in years.

ExampleO What is the total amount for an

investment of $500 invested at 5.25% for 40 years and compounded continuously?

ExampleO What is the total amount for an

investment of $100 invested at 3.5% for 8 years and compounded continuously?

Student guided practice

O Do problem 11 in your book page 278

Half life formulaO The half-life of a substance is the

time it takes for half of the substance to breakdown or convert to another substance during the process of decay. Natural decay is modeled by the function below.

ExampleO Pluonium-239 (Pu-239) has a

half-life of 24,110 years. How long does it take for a 1 g sample of Pu-239 to decay to 0.1 g?

ExampleO Determine how long it will take

for 650 mg of a sample of chromium-51 which has a half-life of about 28 days to decay to 200 mg.

Student guided practice

O Do problem 12 in your book page 278

HomeworkO Do odd problems from 13 to 22 in

your book page 278

Closure O Today we learned about the natural

base of eO Next class were are going to learn

about transforming exponential and logarithmic function