8-6 Natural Logarithms.

Slides:

Advertisements

Similar presentations

Section 5.4 – Properties of Logarithms. Simplify:

Advertisements

Essential Question: What are some of the similarities and differences between natural and common logarithms.

CH. 8.6 Natural Logarithms. Write 2 ln 12 – ln 9 as a single natural logarithm. 2 ln 12 – ln 9 = ln 12 2 – ln 9Power Property = lnQuotient Property 12.

LOGS EQUAL THE The inverse of an exponential function is a logarithmic function. Logarithmic Function x = log a y read: “x equals log base a of y”

Algebra 2 Section 8-6 Daily Goals: ·To understand and use the inverse function of the Exponential Function. ·To understand and know how to apply Natural.

5.4 Exponential and Logarithmic Equations Essential Questions: How do we solve exponential and logarithmic equations?

Exponential and Logarithmic Equations

Functions and Logarithms. One-to-One Functions A function f(x) is one-to-one if f(a) ≠ f(b) whenever a ≠ b. Must pass horizontal line test. Not one-to-one.

7-5 Logarithmic & Exponential Equations

8.6 Natural Logarithms. Natural Logs and “e” Start by graphing y=e x The function y=e x has an inverse called the Natural Logarithmic Function. Y=ln x.

LAWS OF LOGARITHMS SECTION 5.6. Why do we need the Laws? To condense and expand logarithms: To Simplify!

11.3 – Exponential and Logarithmic Equations. CHANGE OF BASE FORMULA Ex: Rewrite log 5 15 using the change of base formula.

8.5 – Exponential and Logarithmic Equations. CHANGE OF BASE FORMULA where M, b, and c are positive numbers and b, c do not equal one. Ex: Rewrite log.

Unit 5: Modeling with Exponential & Logarithmic Functions Ms. C. Taylor.

Natural Logarithms.

Aim: How do we solve exponential equations using common or natural logarithms? Do Now: 1. Solve for x: 3 x = Solve for x: 4 x = 8 3. Solve for x:

Copyright © 2009 Pearson Education, Inc. CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3.

7-6 Solving Natural Log Equations

Warm Up 2. (3 –2 )(3 5 ) (2 6 )(2 8 ) (7 3 ) Simplify. Write in exponential form. x 0 = 1 6. log x x = 1 x 1 = x 7. 0 =

Aim: Exponential Equations using Logs Course: Alg. 2 & Trig. Aim: How do we solve exponential equations using logarithms? Do Now:

Algebra 2 Write 2 ln 12 – ln 9 as a single natural logarithm. 2 ln 12 – ln 9 = ln 12 2 – ln 9Power Property = lnQuotient Property = ln 16Simplify.

SOLVING LOGARITHMIC EQUATIONS Objective: solve equations with a “log” in them using properties of logarithms How are log properties use to solve for unknown.

Daily Warm-UP Quiz 1.Expand: ln x -5 y 2 2x 2. Condense: ½ ln4 + 2 (ln6-ln2) 3. Use properties of logs to solve for x: a. log 81 = x log3 b. log x 8/64.

4.7 (Green) Solve Exponential and Logarithmic Equations No School: Monday Logarithms Test: 1/21/10 (Thursday)

Property of Logarithms If x > 0, y > 0, a > 0, and a ≠ 1, then x = y if and only if log a x = log a y.

Properties of Logarithms and Common Logarithms Sec 10.3 & 10.4 pg

3.3 Logarithmic Functions and Their Graphs

Solving Equations Exponential Logarithmic Applications.

Solving Logarithmic Equations I.. Relationship between Exponential and Logarithmic Equations. A) Logs and Exponentials are INVERSES of each other. 1) That.

LOGARITHMIC AND EXPONENTIAL EQUATIONS Intro to logarithms and solving exponential equations.

8.6 Natural Logarithms.

Natural Logarithms. What you’ll learn To evaluate and simplify natural logarithmic expressions. To solve equations using natural logarithms. Vocabulary.

Review of Logarithms. Review of Inverse Functions Find the inverse function of f(x) = 3x – 4. Find the inverse function of f(x) = (x – 3) Steps.

For b > 0 and b  1, if b x = b y, then x = y.

CHAPTER 5: Exponential and Logarithmic Functions

8.5 – Exponential and Logarithmic Equations

Exponential and Logarithmic Functions

Ch. 8.5 Exponential and Logarithmic Equations

Welcome to Precalculus!

Natural Logarithms.

Solving Exponential and Logarithmic Equations

Exponential Equations

8.5 – Exponential and Logarithmic Equations

3.4 Quick Review Express In 56 in terms of ln 2 and ln 7.

Exponential and Logarithmic Equations

8.6 Solving Exponential & Logarithmic Equations

Copyright © Cengage Learning. All rights reserved.

Ch 3.3: Properties of Logarithms

Logarithmic Functions and Their Graphs

Logarithms and Logarithmic Functions

Essential Question: How do I graph & solve exponential and logarithmic functions? Daily Question: What are the properties of logarithms and how do I use.

Logarithmic and exponential equations

Chapter 10.5 Base e and Natural Logarithms Standard & Honors

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

4.1/4.2 – Exponential and Logarithmic Functions

Copyright © Cengage Learning. All rights reserved.

Natural Logarithms.

Exponential and Logarithmic Functions

3.4 Exponential and Logarithmic Equations

6.3 Logarithms and Logarithmic Functions

LEARNING GOALS – LESSON 7.6

Exponential and Logarithmic Functions

For b > 0 and b ≠ 1, if b x = b y, then x = y.

C2D8 Bellwork: Fill in the table Fill in the blanks on the worksheet

Compound Interest If a principal P is invested at an interest rate r for a period of t years, then the amount A of the investment is given by A = P(1 +

Logarithmic and exponential equations

U6D12 Have out: Bellwork: Fill in the table

Exponential and Logarithmic Functions

Solving Logarithmic Equations

Presentation transcript:

8-6 Natural Logarithms

Objectives Natural Logarithms Natural Logarithmic & Exponential Equations

the x comes down from the exponent. Vocabulary y = ex and y = ln x are inverses So, e and ln are inverse operations. Ex. ln x = 4 ex = 12 elnx = e4 ln (ex) = ln 12 x = e4 x = ln 12 x = 54.6 x = 2.48 Cancel each other the x comes down from the exponent.

Simplify Natural Logarithms Write 2 ln 12 – ln 9 as a single natural logarithm. 2 ln 12 – ln 9 = ln 122 – ln 9 Power Property = ln Quotient Property 122 9 = ln 16 Simplify.

Real World Example Find the velocity of a spacecraft whose booster rocket has a mass ratio 22, an exhaust velocity of 2.3 km/s, and a firing time of 50 s. Can the spacecraft achieve a stable orbit 300 km above Earth? Let R = 22, c = 2.3, and t = 50. Find v. v = –0.0098t + c ln R Use the formula. = –0.0098(50) + 2.3 ln 22 Substitute. –0.49 + 2.3(3.091) Use a calculator. 6.62 Simplify. The velocity is 6.6 km/s is less than the 7.7 km/s needed for a stable orbit. Therefore, the spacecraft cannot achieve a stable orbit at 300 km above Earth.

Solving a Natural Logarithm Equation Solve ln (2x – 4)3 = 6. ln (2x – 4)3 = 6 3 ln (2x – 4) = 6 Power Property ln (2x – 4) = 2 Divide each side by 3. 2x – 4 = e2 Rewrite in exponential form. x = Solve for x. e2 + 4 2 x 5.69 Use a calculator. Check: ln (2 • 5.69 – 4)3 6 ln 401.95 6 5.996 6

Solving an Exponential Equation Use natural logarithms to solve 4e3x + 1.2 = 14. 4e3x + 1.2 = 14 4e3x = 12.8 Subtract 1.2 from each side. e3x = 3.2 Divide each side by 4. ln e3x = ln 3.2 Take the natural logarithm of each side. 3x = ln 3.2 Simplify. x = Solve for x. ln 3.2 3 x 0.388

Real World Example An initial investment of $200 is now valued at $254.25. The interest rate is 6%, compounded continuously. How long has the money been invested? A = Pert Continuously compounded interest formula. 254.25 = 200e0.06t Substitute 254.25 for A, 200 for P, and 0.06 for r. 1.27125 = e0.06t Divide each side by 200. ln 1.27125 = ln e0.06t Take the natural logarithm of each side. ln 1.27125 = 0.06t Simplify. = t Solve for t. ln 1.27125 0.06 4 t Use a calculator. The money has been invested for 4 years.

Homework 8-6 p 472 1,2,10,11,14,15,23,24