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Section 12.6

Section 12.6. Exponential and Logarithmic Equations. Phong Chau. Properties. Power Rule:. Exponential Equations. Equations with variables in the exponents are called Exponential Equations For simple equation, use the followingprinciple :. Solving simple equations.

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Section 12.6

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  1. Section 12.6 Exponential and Logarithmic Equations Phong Chau

  2. Properties • Power Rule:

  3. Exponential Equations • Equations with variables in the exponents are called Exponential Equations • For simple equation, use the followingprinciple:

  4. Solving simple equations These exponential equations are simple because we can express both sides of the equations as a power of the same base

  5. Solving exponential equation Take the natural logarithm on both sides Use Power Rule Divide both sides by ln 2 This is the exact solution Use calculator to find the approximate solution

  6. Strategies for Solving Exponential Equations • Isolate the exponential term • Take the natural logarithm on both sides • Use the power rule to pull the x out of the exponent • Solve the resulting equation • Check the answer in the original equation.

  7. Example Solve: e1.32t– 2000 =0 Solution e1.32t = 2000 We have: Take the natural logarithm ln e1.32t = ln 2000 1.32t = ln 2000 t = (ln 2000)/1.32

  8. Example Solve: 3x +1– 43 = 0 Solution 3x +1 = 43 We have log 3x +1 = log 43 (x +1)log 3= log 43 Power rule for logs x +1= log 43/log 3 x = (log 43/log 3) – 1 The solution is (log 43/log 3) – 1, or approximately 2.4236.

  9. Example

  10. http://www.intmath.com/Exponential-logarithmic-functions/6_Logarithm-exponential-eqns.phphttp://www.intmath.com/Exponential-logarithmic-functions/6_Logarithm-exponential-eqns.php The above link is a good website to learn new concept. It has many applications.

  11. World Population • The world population in billions at time t, where t = 0 represents the year 2000, is given by: • When will the population reach 12 billions?

  12. Strategies for solving Logarithmic Equations • Move all logarithms to left hand side (LHS) • Write the LHS as a single logarithm • Rewrite the equation in exponential form • Solve the resulting equation • Check the answer in the original equation.

  13. Example Solve: log2(6x + 5) = 4. Solution log2(6x + 5) = 4 6x + 5 = 24 6x + 5 = 16 6x = 11 x = 11/6 Check the solution!

  14. Example Solve: log x + log(x + 9) = 1. Solution log x + log(x + 9) = 1 log[x(x + 9)] = 1 x(x + 9) = 101 x2+ 9x = 10 x2+ 9x – 10 = 0 (x – 1)(x + 10) = 0 x – 1 = 0 or x + 10 = 0 x = 1 or x = –10

  15. Check x = 1: log 1 + log(1 + 9) 0 + log(10) 0 + 1 = 1 TRUE x = –10: log (–10) + log(–10 + 9) FALSE The logarithm of a negative number is undefined. The solution is 1.

  16. Example Solve: log3(2x+ 3) – log3(x – 1) = 2. Solution log3(2x + 3) – log3(x – 1) = 2 log3[(2x + 3)/(x – 1)] = 2 (2x + 3)/(x – 1) = 32 (2x + 3)/(x – 1) = 9 (2x + 3) = 9(x – 1) 2x + 3 = 9x – 9 x = 12/7 Check the solution!

  17. Examples

  18. Examples

  19. Group Exercise

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