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8.6 solving exponential and log equations

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8.6 solving exponential and log equations

  1. 1. Solving Exponential Equations If two powers with the same base are equal, then their exponents must be equal. To solve using this property:  Rewrite equation so the bases match.  Set exponents equal to each other.  Solve for x.
  2. 2. Examples: Solve:
  3. 3. Your Turn! Solve
  4. 4. Solving by Taking Logs If it is not easy/possible to rewrite using same bases, take the log (with base b) of both sides. Examples:
  5. 5. Your Turn! Solve
  6. 6. More Examples: Taking Logs Not all equations will be that basic!  Must get the exponential part by itself, first.  Then, take the log of both sides.  Last, continue solving for x. Example:
  7. 7. Example: Solve
  8. 8. Your Turn! Solve
  9. 9. Natural Base Exp. Equations Get the exponential part alone. Take the ln of each side. Example:
  10. 10. Exponential Models Newton’s Law of Cooling – the temperature T of a cooling substance is modeled by where: t = time (in minutes) T0 = initial temp. of substance TR = room temp. r = cooling rate
  11. 11. Example: Your mom is making soup for dinner. When she takes it off the stove, it is 212ºF. The room temp. is 70ºF and the cooling rate of the soup is r = 0.046. How long will you have to wait to eat the soup at a safe temp. of 100ºF?
  12. 12. Solving Log Equations For two logs with the same base: Example: Solve
  13. 13. Your Turn! Solve
  14. 14. Equations with One Log “Chop off the log” and write in exponential form. Then solve for x. Example: Solve
  15. 15. Your Turn! Solve
  16. 16. Extraneous Solutions The domains of log functions are limited. This sometimes produces extraneous solutions. Always check answers in original equation. Example: Solve
  17. 17. Example: Solve
  18. 18. Your Turn! Solve