2. Properties of Logarithms Section 4.3 JMerrill, 2005 Revised, 2008

3. Rules of Logarithms If M and N are positive real numbers and b is ≠ 1: TTTThe Product Rule: llllogbMN = logbM + logbN (The logarithm of a product is the sum of the logarithms) EEEExample: log4(7 9) = log47 + log49 EEEExample: log (10x) = log10 + log x

4. Rules of Logarithms If M and N are positive real numbers and b ≠ 1: TTTThe Product Rule: llllogbMN = logbM + logbN (The logarithm of a product is the sum of the logarithms) EEEExample: log4(7 9) = log47 + log49 EEEExample: log (10x) = log10 + log x YYYYou do: log8(13 9) = YYYYou do: log7(1000x) = log 8 13 + log 8 9 log 7 1000 + log 7 x

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

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

7. Rules of Logarithms If M and N are positive real numbers, b ≠ 1, and p is any real number: TTTThe Power Rule: llllogbMp = p logbM (The log of a number with an exponent is the product of the exponent and the log of that number) EEEExample: log x2 = 2 log x EEEExample: ln 74 = 4 ln 7 YYYYou do: log359 = CCCChallenge: 9log 3 5

8. Prerequisite to Solving Equations with Logarithms SSSSimplifying EEEExpanding CCCCondensing

9. Simplifying (using Properties) llllog94 + log96 = log9(4 6) = log924 llllog 146 = 6log 14 YYYYou try: log1636 - log1612 = YYYYou try: log316 + log24 = YYYYou try: log 45 - 2 log 3 = log 16 3 Impossible! log 5

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

11. Expanding

12. Condensing  Condense: Product Rule Power Rule Quotient Rule

13. Condensing  Condense:

14. Bases  Everything we do is in Base 10.  We count by 10’s then start over. We change our numbering every 10 units.  In the past, other bases were used.  In base 5, for example, we count by 5’s and change our numbering every 5 units.  We don’t really use other bases anymore, but since logs are often written in other bases, we must change to base 10 in order to use our calculators.

15. 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.

16. Change of Base Formula  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!