1. 5.0 Properties of Logarithms AB Review for Ch.5

2. Rules of Logarithms If M and N are positive real numbers and b is ≠ 1: The Product Rule: log b MN = log b M + log b N (The logarithm of a product is the sum of the logarithms) Example: log 4 (7 9) = log 4 7 + log 4 9 Example: log (10x) = log10 + log x

3. Rules of Logarithms If M and N are positive real numbers and b ≠ 1: The Product Rule: log b MN = log b M + log b N (The logarithm of a product is the sum of the logarithms) Example: log 4 (7 9) = log 4 7 + log 4 9 Example: log (10x) = log10 + log x You do: log 8 (13 9) = You do: log 7 (1000x) = log 8 13 + log 8 9 log 7 1000 + log 7 x

4. Rules of Logarithms If M and N are positive real numbers and b ≠ 1: The Quotient Rule (The logarithm of a quotient is the difference of the logs) Example:

5. The Quotient Rule (The logarithm of a quotient is the difference of the logs) Example: You do:

6. Rules of Logarithms If M and N are positive real numbers, b ≠ 1, and p is any real number: The Power Rule: log b M p = p log b M (The log of a number with an exponent is the product of the exponent and the log of that number) Example: log x 2 = 2 log x Example: ln 7 4 = 4 ln 7 You do: log 3 5 9 = 9log 3 5

7. Simplifying (using Properties) log 9 4 + log 9 6 = log 9 (4 6) = log 9 24 log 14 6 = 6log 14 a You try: log 16 36 - log 16 12 = You try: log 3 16 + log 2 4 = You try: log 45 - 2 log 3 = log 16 3 Impossible! log 5

8. Using Properties to Expand Logarithmic Expressions Expand: Use exponential notation Use the product rule Use the power rule

9. Expand:

10. Condense:

11. Change of Base Examine the following problems: log 4 64 = x » we know that x = 3 because 4 3 = 64, and the base of this logarithm is 4 log 100 = x – If no base is written, it is assumed to be base 10 » We know that x = 2 because 10 2 = 100 But because calculators are written in base 10, we must change the base to base 10 in order to use them.

12. Change of Base Formula log 5 8 = Example log 5 8 = This is also how you graph in another base. Enter y 1 =log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10!

13. Find the domain, vertical asymptotes, and x-intercept. Sketch a graph. 13 y x 4 –4 Graphing logarithmic functions

14. Find the domain, vertical asymptotes, and x-intercept. Sketch a graph. y x 4 –4 Graphing logarithmic functions.

15. Homework: MMM pg. 186-188