Chapter 5 Exponential and Logarithmic Functions

1. Chapter 5 Exponential and Logarithmic Functions

2. Logarithmic Functions and Models 5.4 Evaluate the common logarithmic function Evaluate logarithms with other bases Solve basic exponential and logarithmic equations Solve general exponential and logarithmic equations Convert between exponential and logarithmic forms

3. Common Logarithm The common logarithm of a positive number x, denoted log x, is defined by log x= k if and only if x= 10k, where k is a real number. The function given by f(x)= log x is called the common logarithmic function.

4. Common Logarithm If a positive number x can be written as 10k,then log x= k.A logarithm is an exponent k.

5. Common Logarithmic Function Graph The y-axis is a vertical asymptote. The common logarithmic function is one- to-one and always increasing. Its domain is all positive real numbers, and its range is all real numbers.

6. Evaluate • log10 • log 100 • log 1000 • log 10000 • log (1/10) • log (1/100) • log (1/1000) • log 1 • 1 because 101= 10 • 2 because 102= 100 • 3 because 103= 1000 • 4 because 104= 10000 • –1 because 10-1= 1/10 • –2 because 10-2= 1/100 • –3 because 10-3= 1/1000 • 0 because 100= 1

7. Simplify each logarithm by hand. Solution Example: Evaluating common logarithms

8. Graphs of f(x) = 10x and f –1(x) = log x Because f(x) = 10x and f –1(x) = log x represent inverse functions, it follows that their graphs are reflections across the line y = x.

9. Inverse Properties of the Common Logarithm The following inverse properties hold for the common logarithm. log10x = x for all real numbers x and 10logx= x for any positive number x

10. Base-2 Logarithm

11. Base-e Logarithm - Natural Logarithm

12. Logarithm The logarithm with a base a of a positive number x, denoted logax, is defined by logax= k if and only if x= ak, where k is a real number. The function given by f(x)= logax is called the common logarithmic function with base a.

13. Domain and Range of Logarithm The domain of base-a logarithm is (0, ∞). The range of base-a logarithm is (–∞, ∞).

14. Inverse Properties of the Logarithm The following inverse properties hold for logarithms with base a. for all real numbers x and for any positive number x

15. Evaluate each logarithm. Solution Example: Evaluating logarithms

16. Use inverse properties to evaluate each expression. Solution Example: Applying inverse properties

17. Inverses and Graphs The graph of y= logaxis a reflection of the graph of y=axacross the liney= x.

18. Solve Solution Example: Solving a base-10 exponential equation

19. Solve Solution Example: Solving a common logarithmic equation

20. Solve each equation. Solution Example: Solving exponential equations

21. Solve each equation. Solution Example: Solving logarithmic equations