Name : ______________ ( ) Class : ________ Date :_________

1. Name : ______________ ( ) Class : ________ Date :_________ Unit 7: Logarithmic and Exponential Functions Objectives: Graphs Logarithms Common and Natural Logarithms Laws of Logarithms Logarithmic Equations Solving Equations of the Form

2. Graphs of Exponential Functions

3. Graphs of Exponential Functions

4. Graphs of Common Logarithms Graph of y = lg x

5. Logarithms Convert to logarithmic form. The logarithm or index for the given base is -2. The base is 3. Convert to index form. The logarithm or index for the given base is 3. The base is 4.

6. Logarithms Special Cases If a logarithm is defined for base a,then and

7. Logarithms Example 1: Evaluate the following.

8. Logarithms A common logarithm is a logarithm to the base 10. On a scientific calculator, common logarithms can be evaluated using the LOG key. Tables of common logarithms were often used for calculating in the days before the electronic calculator.

9. Logarithms A natural logarithm is a logarithm to the base e. On a scientific calculator, natural logarithms can be evaluated using the LN key. Natural logarithms are also known as Naperian logarithms after John Napier (1550 - 1617).

10. Logarithms From the definitions of logarithms, the following statements are equivalent. and Let’s use these definitions in some examples.

11. Logarithms Example 2: Convert the following to index form. Index form Index form Convert the following to logarithmic form. Logarithmic form Logarithmic form

12. Surds, Indices and Logarithms Example 3: Find y in terms of x. (a) Index form Rearrange (b) Rearrange Index form Alternate form

13. Logarithms Example 4: Solve for x. Evaluate using the calculator. Index form Evaluate and solve for x. In most calculators, the function ex is on the same key as LN.

14. Logarithms Example 5: Solve for x. Evaluate using the calculator. Index form Evaluate and solve for x. In most calculators, the function 10x is on the same key as LOG.

15. Logarithms The Power Law The Product Law The Quotient Law

16. Logarithms The Change of Base Law Let’s use these laws in some examples. A special case

17. Logarithms Example 6: Evaluate the following. (a) Apply the power law. Apply the power law. Combine using the product and quotient laws. (b) Combine using the product and quotient laws.

18. Logarithms Example 7: Separate using the product and quotient laws. Apply the power law.

19. Logarithms Example 8: Find y in terms of x. Arrange the log terms on one side. Combine, applying the quotient law. Index form Rearrange and solve the equation.

20. Logarithms Example 9: Evaluate the following. Apply the change of base law. Express as powers of 2, 5 and 10. Apply the power law.

21. Logarithms An Important Property of Logarithms For two logarithms of the same base, Let’s use this property to solve some logarithmic equations.

22. Logarithms Example : 10 Combine using the product law. Use the property of logarithms. Remember to check if the results are acceptable. So, x = 3.

23. Logarithms Example 11: Solve the following equation. Apply the power law. Remember to check if the results are acceptable. Index form log4(6 – x) is defined for x = –122.

24. Logarithms Example 12 : Solve the following equation. Apply the change of base law. Substitute Both results are acceptable.

25. Logarithms An Important Property of Logarithms For two logarithms of the same base, Let’s use this property to solve some logarithmic equations.

26. Logarithms Example 13 : Combine using the product law. Use the property of logarithms. Remember to check if the results are acceptable. So, x = 3.

27. Logarithms Example 14 : Solve the following equation. Apply the power law. Remember to check if the results are acceptable. Index form log4(6 – x) is defined for x = –122.

28. Logarithms Example 15 : Solve the following equation. Apply the change of base law. Substitute Both results are acceptable.