1. Chapter 5: Inverse, Exponential, and Logarithmic Functions
5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions.

2. 5.5 Exponential and Logarithmic Equations and Inequalities (2 of 2)
Type I Exponential Equations
Solved in Section 5.2
Easily written as powers of same base
i.e. 125x = 5x
Type 2 Exponential Equations
Cannot be easily written as powers of same base
i.e. 7x = 12
General strategy: take the logarithm of both sides and apply the power rule to eliminate variable exponents.

3. 5.5 Type 2 Exponential Equations
Example Solve 7x = 12.

4. 5.5 Type 2 Exponential Equations
Example Solve 7x = 12.

5. Solving a Type 2 Exponential Inequality
Example Solve 7x < 12. Solution From the previous example, 7x = 12 when x  Using the graph below, y1 = 7x is below the graph y2 = 12 for all x-values less than

6. 5.5 Solving a Type 2 Exponential Equation

7. 5.5 Solving a Type 2 Exponential Equation

8. Solving a Logarithmic Equation of the Type log x = log y
Example Solve

9. Solving a Logarithmic Equation of the Type log x = log y
Example Solve Analytic Solution The domain must satisfy x + 6 > 0, x + 2 > 0, and x > 0. The intersection of these is (0, ).

10. Solving a Logarithmic Equation of the Type log x = log y
Since the domain of the original equation was (0, ), x = –3 cannot be a solution. The solution set is {2}.

11. Solving a Logarithmic Equation of the Type log x = log y
The point of intersection is at x = 2. Notice that the graphs do not intersect at x = –3, thus supporting our conclusion that –3 is an extraneous solution.

12. Solving a Logarithmic Equation of the Type log x = k

13. Solving a Logarithmic Equation of the Type log x = k

14. Solving Equations Involving both Exponentials and Logarithms

15. Solving Equations Involving both Exponentials and Logarithms

16. 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions
t represents time
k > 0 represents the growth constant, and k < 0 represents the decay constant

17. Finding the Age of a Fossil
Example Carbon 14 is a radioactive form of carbon found in all living plants and animals. After a plant or animal dies, the radiocarbon disintegrates. Scientists determine the age of the remains by comparing the amount of carbon 14 present with the amount found in living plants and animals. The amount of carbon 14 present after t years is given by

18. Finding the Age of a Fossil
Example Carbon 14 is a radioactive form of carbon found in all living plants and animals. After a plant or animal dies, the radiocarbon disintegrates. Scientists determine the age of the remains by comparing the amount of carbon 14 present with the amount found in living plants and animals. The amount of carbon 14 present after t years is given by

19. 5.6 Finding the Age of a Fossil (2 of 2)
The age of the fossil is about 13,000 years.

20. Finding Half-life (there are more ways to do this)
Example Links posted on Twitter generally experience half of their hits H within the first 2.8 hours. This pattern continues over time so that the number of hits on a link decreases by half over each 2.8-hour period. Hits on a Twitter link decay according to the function H(t) = H0e-kt, where t is time in hours. Find the exact value of k and then approximate k. Solution The half-life tells us that when t = 2.8, A(t) = (½)H0.

21. Finding Half-life

22. Financial Applications
Example How long will it take $1000 invested at 6% interest compounded quarterly to grow to $2700?

23. Financial Applications
Example How long will it take $1000 invested at 6% interest compounded quarterly to grow to $2700?
Solution Find t when A = 2700, P = 1000, r = 0.06,
and n = 4.

24. Financial Applications

25. Financial Applications