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Pauline Cha Amanda Jocson Stephanie Yu 3rd Period

Secs. 8.4 - 8.6. Pauline Cha Amanda Jocson Stephanie Yu 3rd Period. 8.4 Concepts. In order to solve logarithms , you must know all the properties: product, quotient, and power property. 8.5 Concepts. To evaluate a logarithm with any base , you can use the Change of Base Formula .

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Pauline Cha Amanda Jocson Stephanie Yu 3rd Period

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  1. Secs. 8.4 - 8.6 Pauline Cha Amanda Jocson Stephanie Yu 3rd Period

  2. 8.4 Concepts • In order to solve logarithms, you must know all the properties: product, quotient, and power property.

  3. 8.5 Concepts • To evaluate a logarithm with any base, you can use the Change of Base Formula. • Exponential equation is an equation of the form bcx = a, where the exponent includes a variable, and you can solve one by taking the logarithm of each side of the equation. • You can also solve an exponential equation by using the Change of Base Formula. • Apply the properties of logarithms to simplify expressions when solving logarithm equations.

  4. 8.6 Concepts • The inverse of function y = ex is the natural logarithmic function, which is written as ln y = x.  • Properties of logs apply to natural logs also, which you can use to solve equations. • You can use natural logs to solve investing problems utilizing the continuously compounded interest formulaA = Pert, where A = amount in account, P = principal (initial investment), r = annual rate of interest, and t = time in yrs.

  5. 8.4 Examples • Properties of Logs • 4 log2W + log2X Power and Product Property = log2(W4 X) logbMN = logbM + logbN logb M/N = logbM - logbN logbMx = xlogbM • Expanding Logs • log10 qr Quotient Property = log10q - log10r • log7r5 Product Property = log 7 + log r5 Power Property = log 7 + 5 log r

  6. 8.5 Examples • Solve Exponential Equations • Solve 82x = 40 log 82x = log 40 Common log on each side 2x log 8 = log 40 Power Property x = log 40  2 log 8 Divide each side by 2 log 8 x = about .887 Usecalculator • 82(.887) = 40.002 = about 40 

  7. 8.5 Examples • Change of Base Formula Evaluate log15254 then convert to log base 5 • Log15254 = log 254  log15 = 2.045 COB Formula log15254 = log5x Write equation 2.045 = log5x Sub log15254 for 2.045 2.045 = log x  log5COB log5 2.045 = log x Multiply each side by log5 1.429 = log x Simplify x = 101.429 Write inexponential form = 26.853

  8. 8.5 Examples • Solve a Logarithmic Equation • Solve 2 log x - log 5 = 3 log (x2  5) = 3 Write as a single log (x2  5) = 103 Write in exponential form x2 = 5(1000) Multiply each side by 5 x = ± 50 2 x =  50 2Solution is positive

  9. 8.6 Examples • Solve Natural Log Equation • Solve ln (4x + 6)2 = 12 (4x + 6)2 = e12 Exponential form (4x + 6)2 = 162745.8Use calculator 4x + 6 =162745.8Square root of each side 4x + 6 =  403.418 Solve for x = 99.35

  10. 8.6 Examples • Compounded Continuously An initial investment of $200 is now valued at $250.30. The interest rate is 7%, compounded continuously. How long has the money been invested? A = Pert 250.30 = 200e0.07t ln 1.2515 = ln e0.07t ln 1.2515 = 0.07t ln 1.2515  .07 = t t = about 3.20 yrs.

  11. Worksheet Answers 8.4- 8.5- 8.6- 1. log x⁷/y2 1. y = 7.90 1. ln 192 2. log₅8 + 1/2log₅x 2. x = 16.98 2. x= 48.8 3. logm + 1/2log9 - 2logs 3. x = 667 3. x = .61 4. C 4. B 4. D 5. A 5. C 5. B

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