1 / 19

Understanding Exponential Functions and Natural Logarithms

Learn how to solve equations involving e or natural logarithms, simplify expressions using ln and e, and apply continuous compounding interest formulas. Explore the value of e and its properties. Practice with examples and a quiz.

fking
Download Presentation

Understanding Exponential Functions and Natural Logarithms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Natural Base, e 4-6 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

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

  3. Objectives Use the number e to write exponential functions. Solve equations involving e or natural logarithms.

  4. Essential Question How do you solve equations involving e or natural logarithms?

  5. Vocabulary natural logarithm natural logarithmic function

  6. Recall the compound interest formulaA = P(1 + )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. r 1 n n 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 + )n.

  7. 1 n As n gets very large, interest is continuouslycompounded. Examine the graph of f(n)= (1 + )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.

  8. Caution The decimal value of e looks like it repeats: 2.718281828… The value is actually 2.71828182890… There is no repeating portion.

  9. A logarithm with a base of e is called a natural logarithmand is abbreviated as “ln” (rather than as loge). Natural logarithms have the same properties as log base 10 and logarithms with other bases. The natural logarithmic functionf(x) = ln x is the inverse of the natural exponential function f(x) = ex.

  10. You use a calculator to evaluate a ln (natural log) expression. The ln button is found on the left side of your calculator below the log button.

  11. Evaluate ln 234. Press ln (234) Press enter. ln 234 ≈ 5.4553 to four decimal places.

  12. Example 2: Simplifying Expression with e or ln Simplify. Remember: ln and base e are inverses. They cancel each other out. A. ln e0.15t B. e3ln(x +1) ln e0.15t= 0.15t e3ln(x +1) = (x + 1)3 C. ln e2x + ln ex ln e2x+ ln ex = 2x + x = 3x

  13. Check It Out! Example 2 Simplify. a. ln e3.2 b. e2lnx ln e3.2= 3.2 e2lnx= x2 c.ln ex +4y ln ex + 4y= x + 4y

  14. 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.

  15. Example 3: Economics Application What is the total amount for an investment of $500 invested at 5.25% for 40 years and compounded continuously? A = Pert A = 500e0.0525(40) Substitute 500 for P, 0.0525 for r, and 40 for t. A ≈ 4083.08 Use the ex key on a calculator. The total amount is $4083.08.

  16. Check It Out! Example 3 What is the total amount for an investment of $100 invested at 3.5% for 8 years and compounded continuously? A = Pert A = 100e0.035(8) Substitute 100 for P, 0.035 for r, and 8 for t. A ≈ 132.31 Use the ex key on a calculator. The total amount is $132.31.

  17. Lesson Quiz Simplify. 1. ln e–10t 2.e0.25 lnt –10t t0.25 3. –ln ex –x 4. 2ln ex2 2x2 5. What is the total amount for an investment of $1000 invested at 7.25% for 15 years and compounded continuously? ≈ $2966.85

More Related