1. Section 6.3 – Exponential Functions Laws of Exponents If s, t, a, and b are real numbers where a > 0 and b > 0, then: Definition: “a” is a positive real number and does not equal 1 “C” is a real number and does not equal 0 The domain of f(x) is the set of all real numbers “a” is the base and is the Growth Factor

2. Section 6.3 – Exponential Functions Examples

3. Section 6.3 – Exponential Functions Examples

4. Section 6.3 – Exponential Functions The domain is the set of all real numbers. The range is the set of all positive real numbers. The y-intercept is 1; x-intercepts do not exist. The x-axis (y = 0) is a horizontal asymptote, as x . If a > 1, the f(x) is increasing function. The graph contains the points (0, 1), (1, a), and (-1, 1/a). The graph is smooth and continuous.

5. Section 6.3 – Exponential Functions The graph of the exponential function is shown below. a

6. Section 6.3 – Exponential Functions The domain is the set of all real numbers. The range is the set of all positive real numbers. The y-intercept is 1; x-intercepts do not exist. The x-axis (y = 0) is a horizontal asymptote, as x . If 0 < a < 1, then f(x) is a decreasing function. The graph contains the points (0, 1), (1, a), and (-1, 1/a). The graph is smooth and continuous.

7. Section 6.3 – Exponential Functions The graph of the exponential function is shown below.

8. Section 6.3 – Exponential Functions Euler’s Constant – e The value of the following expression approaches e, as n approaches . Using calculus notation, Growth and decay Compound interest Differential and Integral calculus with exponential functions Applications of e Infinite series

9. Section 6.3 – Exponential Functions Solving Exponential Equations Theorem

10. Section 6.3 – Exponential Functions Solving Exponential Equations

11. Section 6.4 – Logarithmic Functions The exponential and logarithmic functions are inverses of each other. The logarithmic function is defined by The x-intercept is 1 and the y-intercept does not exist. The y-axis (x = 0) is a vertical asymptote. If 0 < a < 1, then the logarithmic function is a decreasing function. The graph contains the points (1, 0), (a, 1), and (1/a, –1). The graph is smooth and continuous. If a > 1, then the logarithmic function is an increasing function.

12. Section 6.4 – Logarithmic Functions The graph of the logarithmic function is shown below. The natural logarithmic function The common logarithmic function

13. Section 6.4 – Logarithmic Functions a

14. a

16. a 

17. a 

18. a 

19. a 

20. Change the exponential statements to logarithmic statements Change the logarithmic statements to exponential statements Solve the following equations

21. Section 6.4 – Logarithmic Functions Solve the following equations