1. 5.6 Exponential and Logarithmic Equations
Solve exponential equations
Solve logarithmic equations
Solve exponential and logarithmic inequalities

2. Exponential Equation
An equation in which one or more variables occur in the exponent of an expression is called an exponential equation.

3. Example: Modeling world population (1 of 3)

4. Example: Modeling world population (2 of 3)

5. Example: Modeling world population (3 of 3)
This model predicts that world population might reach 8 billion during 2024.

6. Example: Modeling the decline of bluefin tuna (1 of 4)
Bluefin tuna are large fish that can weigh 1500 pounds and swim at speeds of 55 miles per hour. Because they are used for sushi, a prime fish can be worth over $50,000. As a result, the western Atlantic bluefin tuna have had their numbers decline exponentially.
(Source: B. Freedman, Environmental Ecology.)

7. Example: Modeling the decline of bluefin tuna (2 of 4)
a. Estimate the number of bluefin tuna in 1974 and b. Determine symbolically the year when they numbered 50 thousand.

8. Example: Modeling the decline of bluefin tuna (3 of 4)
Solution a. To determine their numbers in 1974 and 1991, evaluate f(0) and f(17).
Bluefin tuna decreased from 230 thousand in 1974 to about 27 thousand in 1991.

9. Example: Modeling the decline of bluefin tuna (4 of 4)
b. Solve the equation f(x) = 50 for x.
They numbered about 50 thousand in  1986.

10. Example: Solving exponential equations symbolically (1 of 4)
Solve each equation.

11. Example: Solving exponential equations symbolically (2 of 4)

12. Example: Solving exponential equations symbolically (3 of 4)

13. Example: Solving exponential equations symbolically (4 of 4)

14. Example: Solving exponential equations graphically

15. Logarithmic Equations
Logarithmic equations contain logarithms. To solve a logarithmic equation, we use the inverse property

16. Example: Solving a logarithmic equation

17. Example: Solving a logarithmic equation symbolically (1 of 5)
In developing countries, there is a relationship between the amount of land a person owns and the average daily calories consumed. This relationship is modeled by the formula C(x) = 280 ln(x + 1) , where x is the amount of land owned in acres and 0 ≤ x ≤ 4. a. Find the average caloric intake for a person who owns no land.

18. Example: Solving a logarithmic equation symbolically (2 of 5)
b. A graph of C is shown. Interpret the graph.
c. Determine symbolically the number of acres owned by someone whose average intake is 2000 calories per day.

19. Example: Solving a logarithmic equation symbolically (3 of 5)
Solution a. Since C(0) = 280 ln (0 + 1) = 1925, a person without land consumes an average of 1925 calories per day.

20. Example: Solving a logarithmic equation symbolically (4 of 5)
b. The y-intercept of (0, 1925) represents the caloric intake of calories for a person who owns no land. As the amount of land x increases, the caloric intake y also increases.
However, the rate of increase slows. This would be expected because there is a limit to the number of calories an average person would eat, regardless of his or her economic status.

21. Example: Solving a logarithmic equation symbolically (5 of 5)
c. Solve the equation C(x) = 2000.
A person who owns about 0.3 acre has an average intake of calories per day.

22. Example: Solving logarithmic equations symbolically (1 of 4)
Solve each equation.

23. Example: Solving logarithmic equations symbolically (2 of 4)
Solution

24. Example: Solving logarithmic equations symbolically (3 of 4)
However, −1 is not a solution since log2x in the given equation is undefined. The only solution is 1.

25. Example: Solving logarithmic equations symbolically (4 of 4)
Substituting x = 0 and x = −4 in the given equation shows that 0 is a solution but −4 is not a solution.

26. Exponential and Logarithmic Inequalities
If the equality sign in an exponential or logarithmic equation is replaced with <, >, ≤, or ≥ an exponential inequality or logarithmic inequality results. Many times it helps to use our understanding of exponential and logarithmic functions and their graphs to solve these types of inequalities. The fact that equality is often the boundary between greater than and less than is helpful in determining the solution set.

27. Example: Solving exponential inequalities
Solve the inequality. Write the solution set in interval notation.