Chapter 4 4-6 The Natural base, e

Quiz#1 Express each as a single logarithm. 1. log 6 9+ log 6 24 Simplify. 3 .log 𝟐 𝟖 𝟏𝟎 4. log 𝟒 𝟒 𝒙−𝟏 5. 𝟏 𝟎 𝒍𝒐𝒈𝟏𝟐𝟓

Quiz#1 Solve. 6. 4 3𝑥−1 = 8 𝑥+1 7. 3 2𝑥−1 = 20 8. lo g7 ( 5x+3 )= 3 6. 4 3𝑥−1 = 8 𝑥+1 7. 3 2𝑥−1 = 20 8. lo g7 ( 5x+3 )= 3 9. log ( 3x + 1 ) - log 4 = 2 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 Base 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 base 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 base 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 functions 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 Graph f(x) = ex–2 + 1. Make a table. Because e is irrational, the table values are rounded to the nearest tenth.

Example 2 Graph f(x) = ex – 3.

Student guided practice Do problems 2-5 in your book page 278

Natural logs A logarithm with a base of e is called a natural logarithm and is abbreviated as “ln” (rather than as loge). Natural logarithms have the same properties as log base 10 and logarithms with other bases.

Natural logs The natural logarithmic function f(x) = ln x is the inverse of the natural exponential function f(x) = ex.

Natural logs The domain of f(x) = ln x is {x|x > 0}. The range of f(x) = ln x is all real numbers All of the properties of logarithms from Lesson 4-4 also apply to natural logarithms.

Example #3 Simplify. A. ln e0.15t B. e3ln(x +1) C. ln e2x + ln ex

Student guided practice Do problems 6-10 in your book page 278

ContinuouslyCompounded 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.

Example What is the total amount for an investment of $500 invested at 5.25% for 40 years and compounded continuously?

Example What is the total amount for an investment of $100 invested at 3.5% for 8 years and compounded continuously?

Student guided practice Do problem 11 in your book page 278

Half life formula 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.

Example 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?

Example 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 Do problem 12 in your book page 278

Homework Do odd problems from 13 to 22 in your book page 278

Closure Today we learned about the natural base of e Next class were are going to learn about transforming exponential and logarithmic function