6.6 The Natural Base, e Objectives: Evaluate natural exponential and natural logarithmic functions.

Slides:

Advertisements

Similar presentations

6.6 The Natural Base, e.

Advertisements

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.

Financial Models (NEW) Section 5.7. Compound Interest Formula If P represents the principal investment, r the annual interest rate (as a decimal), t the.

9.5 Exponential Equations & Inequalities. Logarithmic vocabulary Consider: log 260 Also: log 0.26 Ex 1) Underline the mantissa & circle the characteristic.

The Natural Base, e 7-6 Warm Up Lesson Presentation Lesson Quiz

Logarithmic Functions

Exponential and Logarithmic Functions. Exponential Functions Vocabulary – Exponential Function – Logarithmic Function – Base – Inverse Function – Asymptote.

4.3 Logarithmic Functions and Graphs Do Now Find the inverse of f(x) = 4x^2 - 1.

Pre-Calc Lesson 5-5 Logarithms

Exponential & Logarithmic Functions

Warm Up Simplify. x 1. log 10 x 2. log b b 3w log z 3w3w z 4. b log b (x – 1 ) x – 1.

Logarithms and Exponential Equations Ashley Berens Madison Vaughn Jesse Walker.

Lesson 5-5 Logarithms. Logarithmic functions The inverse of the exponential function.

Chapter The Natural base, e.

6. 3 Logarithmic Functions Objectives: Write equivalent forms for exponential and logarithmic equations. Use the definitions of exponential and logarithmic.

Pre-Calc Lesson 5-7 Exponential Equations; Changing Bases An Exponential Equation is an equation that contains a variable in the exponent. Some exponential.

Notes 6.6, DATE___________

Exponential Equations Like Bases. Warm Up  The following quadratic equation has exactly one solution for x. Find the value of k. Explore more than one.

 If m & n are positive AND m = n, then  Can solve exponential equation by taking logarithm of each side of equation  Only works with base 10.

Exponential Functions An exponential function is of the form f (x) = a x, where a > 0. a is called the base. Ex. Let h(x) = 3.1 x, evaluate h(-1.8).

6.6 The Natural Base, e Objectives: Evaluate natural exponential and

Chapter 11 Section 11.1 – 11.7 Review. Chapter 11.1 – 11.4 Pretest Evaluate each expression 1. (⅔) -4 = ___________2. (27) - ⅔ = __________ 3. (3x 2 y.

Chapter 1.5 Functions and Logarithms. One-to-One Function A function f(x) is one-to-one on a domain D (x-axis) if f(a) ≠ f(b) whenever a≠b Use the Horizontal.

6.6 The Natural Base, e Objectives: Evaluate natural exponential and natural logarithmic functions.

Simplify. 1. log10x 2. logbb3w 3. 10log z 4. blogb(x –1) 5.

Exponential and Logarithmic Functions

4.4 Evaluate Logarithms and Graph Logarithmic Functions Part 2.

5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate,

3.2 Logarithmic Functions 2015 Digital Lesson. 3.1 Warm-up Mr. Smith deposited $6,500 in an account that pays the account pays 4.5% interest, compounded.

The Natural Base, e 4-6 Warm Up Lesson Presentation Lesson Quiz

5.7 – Exponential Equations; Changing Bases

GRAPHING EXPONENTIAL FUNCTIONS f(x) = 2 x 2 > 1 exponential growth 2 24–2 4 6 –4 y x Notice the asymptote: y = 0 Domain: All real, Range: y > 0.

8.2 Properties of Exponential Functions 8.3 Logarithmic Functions as Inverses.

6.2 Exponential Functions Objective: Classify an exponential function as representing exponential growth or exponential decay. Calculate the growth of.

Objectives Use the number e to write and graph exponential functions representing real-world situations. Solve equations and problems involving e or natural.

4.2 Logarithmic Functions

How do we solve exponential and logarithmic equations and equalities?

3.3 Logarithmic Functions and Their Graphs

Objective: Students will be able to write equivalent forms for exponential and logarithmic functions, and can write, evaluate, and graph logarithmic functions.

Unit 8, Lesson 2 Exponential Functions: Compound Interest.

Section 6: The Natural Base, e. U se the number e to write and graph exponential functions representing real-world situations. S olve equations and problems.

Holt McDougal Algebra The Natural Base, e Warm Up Simplify. x 1. log 10 x 2. log b b 3w log z 3w3w z 4. b log b (x – 1 ) x – 1 Slide 1 of 16.

Focus: 1. Recognize and evaluate logarithmic functions. Use logarithmic functions to solve problems.

7-6 The Natural Base, e Entry Task Lesson 7.6 Practice Holt Algebra 2.

3.2 Logarithmic Functions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Intro Solving for an answer Solving for a baseSolving.

LEQ: HOW DO YOU EVALUATE COMMON LOGARITHMS? Common Logarithms Sec. 9-5.

Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate, and.

10.5 Base e and Natural Logarithms Students will be able to… 1) Evaluate expressions involving the natural base and natural logarithms. 2) Solve exponential.

The Natural Base, e 7-6 Warm Up Lesson Presentation Lesson Quiz

Logarithmic Functions

The Natural Base, e Warm Up Lesson Presentation Lesson Quiz

The Natural Base, e 7-6 Warm Up Lesson Presentation Lesson Quiz

6. 3 Logarithmic Functions Objectives: Write equivalent forms for exponential and logarithmic equations. Use the definitions of exponential and logarithmic.

6.6 The Natural Base, e Objectives: Evaluate natural exponential and

5.4 Logarithmic Functions and Models

Logarithmic Functions

Logarithmic Functions and Their Graphs

A function is given by a formula. Determine whether it is one-to-one

The Natural Base, e 4-6 Warm Up Lesson Presentation Lesson Quiz

Exponential Functions

6.3 Logarithms and Logarithmic Functions

The Natural Base, e 7-6 Warm Up Lesson Presentation Lesson Quiz

Logarithmic Functions

The Natural Base, e 4-6 Warm Up Lesson Presentation Lesson Quiz

Warm-Up Evaluate log x for each value. x = 10 x = 0.1 x = -10 x = 1

Goes along with 4.4 (GREEN book)

Logarithmic Functions

4.3 Use Functions Involving e

Logarithmic Functions

The Natural Base, e 7-6 Warm Up Lesson Presentation Lesson Quiz

Presentation transcript:

6.6 The Natural Base, e Objectives: Evaluate natural exponential and natural logarithmic functions.

The natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.

The Natural Exponential Function The exponential function with base e, f(x) = e x is called the natural exponential function and e is called the natural base. The function e x is graphed. Notice that the domain is all real numbers The range is all positive numbers.

Ex 1. Evaluate f(x) = e x to the nearest thousandth for each value of x below. a. x= 2 e 2 = b. x= ½ e 1/2 = c. x = -1 e -1 =.368 d. x = 6 e 6 = e. x = 1/3 e 1/3 = f. x = -2 e -2 =.135

Continuous Compounding Formula

Ex 2: An investment of $1000 earns an annual interest rate of 7.6%. Compare the final amounts after 8 years for interest compounded quarterly and for interest compounded continuously. Quarterly A = P(1+ r/n) nt A = 1000(1+.076/4) 4*8 A = Continuously A = Pe rt A = 1000e.076 * 8 A =

Ex 3: Find the value of $500 after 4 years invested at an annual interest rate of 9% compounded continuously. P = 500 t = 4 r =.09 A = 500e.36 = $716.66

The Natural Logarithmic Function The natural logarithmic function y = log e x, abbreviated y = In x, is the inverse of the natural exponential function, y = e x. The function y = In x is graphed along with y = e x. y=x y=e x y = Inx

Ex 4 Evaluate f(x) = ln x to the nearest thousandth for each value of x below. a.x = 2 ln 2 =.693 b. x = ½ In ½ = c. x = -1 In -1 = undefined d. x = 5 In 5 = e. x= 0.85 In.85 = f. x = 1 In 1 = 0

The natural logarithmic function can be used to solve an equation of the form A = Pe rt for the exponent t in order to find the time it takes for an investment that is compounded continuously to reach a specific amount. **** In e = 1 ****

Ex 5 How long does it take for an investment to double at an annual interest rate of 8.5% compounded continuously?

Ex 5 How long does it take for an investment to triple at an annual interest rate of 7.2% compounded continuously?

► Ex 7 Radiocarbon Dating Suppose that archaeologists find scrolls and claim that they are 2000 years old. Tests indicate that the scrolls contain 78% of their original carbon-14.

Homework Pg #12-30 even #72-75