1. 5.7 – Exponential Equations

2. 5.7 Exponential Equations Objectives: I will be able to…  Solve Exponential Equations using the Change of Base Formula Vocabulary: logarithms, natural logarithms

3. Daily Objectives Perform the Change of Base formula. Master solving tricky logarithm equations. ▫Exponent variables

4. Topic One: Change of Base Formula The BASE goes to the BOTTOM We typically use log base 10 so we can solve on our calculator!

5. Example 1: Using Change of Base Formula This is a great problem to use the change of base formula on – Why? On the other hand, why is problem #2 not the type of problem that you should use the change of base formula on? #3 can work with either – Why?

6. Example 2: Solve for a variable exponent when it’s impossible to get same base 1.Get the base/exponent by itself on one side of the equation (use PEMDAS) 2.Take the log of both sides 3.Bring down the exponent via the Log Power Rule 4.Get x by itself using Change of Base Rule! 5.**Remember: ln(e)=1 and log(10)=1 x ~ 0.81137

7. Example 3: Solve for a variable exponent x ~ 1.6989

8. t ~ 20.352 years Example 4: Solve for a variable exponent (time)

9. Practice time!

10. #1: Solve for a variable exponent x ~ -0.04668

11. #2: Solve for a variable exponent y ~ 2.322

12. #3: Solve for a variable exponent 5 = (1.6) x x ~ 3.4243

13. #4: Solve for a variable exponent x ~ 1.3863

14. #5: Solve for a variable exponent x ~ 0.2792

15. #6: Solve for a variable exponent No solution

16. #7: Solve for a variable exponent The number of bacteria present in a culture N(t) at time t hours is given by N(t)=3000(2) t How long will it take for the population to triple in size? t ~ 1.5850 hours

17. Homework p. 205 #5-14

18. t ~ 6.7578 years Example 7: Solve for a variable exponent (time)

19. Example 8: Solve for a variable exponent A biologist decides that an epidemic spreads through a population of a city according to the following model p(t) = 1 − e −0.34t where p(t) represents that fraction of the city’s population which has come down with the disease, and t is in weeks. How long will it take for 90% of the city to become infected? t ~ 6.7723 weeks

20. Exit Ticket