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8-6 Natural Logarithms

8-6 Natural Logarithms. Objectives. Natural Logarithms Natural Logarithmic & Exponential Equations. Vocabulary. y = e x and y = ln x are inverses So, e and ln are inverse operations. Ex. ln x = 4 e x = 12 e lnx = e 4 ln (e x ) = ln 12

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8-6 Natural Logarithms

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  1. 8-6 Natural Logarithms

  2. Objectives Natural Logarithms Natural Logarithmic & Exponential Equations

  3. 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.

  4. 122 9 = ln Quotient Property 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 16 Simplify.

  5. –0.49 + 2.3(3.091) Use a calculator. 6.62 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 RUse the formula. = –0.0098(50) + 2.3 ln 22Substitute. 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.

  6. 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 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 = e2Rewrite in exponential form.

  7. ln 3.2 3 x = Solve for x. x 0.388 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.

  8. ln 1.27125 0.06 = tSolve for t. 4 tUse a calculator. 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 = PertContinuously compounded interest formula. 254.25 = 200e0.06tSubstitute 254.25 for A, 200 for P, and 0.06 for r. 1.27125 = e0.06tDivide each side by 200. ln 1.27125 = ln e0.06tTake the natural logarithm of each side. ln 1.27125 = 0.06tSimplify. The money has been invested for 4 years.

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

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