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The Natural Base, e. 4-6. Warm Up. Lesson Presentation. Lesson Quiz. Holt McDougal Algebra 2. Holt Algebra 2. Warm Up Simplify. x. 1. log 10 x. 3 w. 2. log b b 3 w. z. 3. 10 log z. x – 1. 4. b log b ( x – 1 ). 3 x – 2. 5. Objectives.

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4-6

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  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

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