1. Honors Algebra 21 Properties of Logarithms During this lesson, you will:  Expand the logarithm of a product, quotient, or power  Simplify (condense) a sum or difference of logarithms

2. Honors Algebra 22 Part 1: Expanding Logarithms

3. Mrs. McConaughyHonors Algebra 23 The Product Rule Let M, N, and b be any positive numbers, such that b ≠ 1. log b (M ∙ N ) = log b M+ log b N The logarithm of a product is the sum of the logarithms. Connection: When we multiply exponents with a common base, we add the exponents. PROPERTY: The Product Rule (Property)

4. 4 Example Expanding a Logarithmic Expression Using Product Rule log (4x) = log 4 + log x The logarithm of a product is The sum of the logarithms. Use the product rule to expand: a.log 4 ( 7 9) = _______________ b. log ( 10x) = ________________ = ________________ log 4 ( 7) + log 4 (9) log ( 10) + log (x) 1 + log (x)

5. Honors Algebra 25 Property: The Quotient Rule (Property) The Quotient Rule Let M, N, and b be any positive numbers, such that b ≠ 1. log b (M / N ) = log b M - log b N The logarithm of a quotient is the difference of the logarithms. Connection: When we divide exponents with a common base, we subtract the exponents.

6. Mrs. McConaughy6 log (x/2) = log x - log 2 Example Expanding a Logarithmic Expression Using Quotient Rule The logarithm of a quotient is The difference of the logarithms. Use the quotient rule to expand: a.log 7 ( 14 /x) = ______________ b. log ( 100/x) = ______________ = ______________ log 7 ( 14) - log 7 (x) log ( 100) - log (x) 2 - log (x)

7. Honors Algebra 27 PROPERTY: The Power Rule (Property) The Power Rule Let M, N, and b be any positive numbers, such that b ≠ 1. log b M x = x log b M When we use the power rule to “pull the exponent to the front,” we say we are _________ the logarithmic expression. expanding

8. Honors Algebra 28 Example Expanding a Logarithmic Expression Using Power Rule Use the power rule to expand: a.log 5 7 4 = _______________ b. log √x = ________________ = ________________ 4log 5 7 log x 1/2 1/2 log x

9. Honors Algebra 29 Summary: Properties for Expanding Logarithmic Expressions Properties of Logarithms Let M, N, and b be any positive numbers, such that b ≠ 1. Product Rule: Quotient Rule: Power Rule: log b (M ∙ N ) = log b M+ log b N log b (M / N ) = log b M - log b N log b M x = x log b M NOTE: In all cases, M > 0 and N >0.

10. Honors Algebra 210 Check Point: Expanding Logarithmic Expressions Use logarithmic properties to expand each expression: a.log b x 2 √y b. log 6 3 √x 36y 4 log b x 2 + log b y 1/2 2log b x + ½ log b y log 6 x 1/3 - log 6 36y 4 log 6 x 1/3 - (log 6 36 + log 6y 4 ) 1/3log 6 x - log 6 36 - 4log 6 y 2

11. Honors Algebra 211 Check Point: Expanding Logs Expand: log 2 3xy 2 log 8 2 6 (xy) 2 = log 2 3 + log 2 x + 2log 2 y = log 8 2 6 + log 8 x 2 + log 8 y 2 = 6log 8 2 + 2log 8 x + 2log 8 y NOTE: You are expanding, not condensing (simplifying) these logs.

12. Honors Algebra 212 Part 2: Condensing (Simplifying) Logarithms

13. Mrs. McConaughyHonors Algebra 213 Part 2: Condensing (Simplifying) Logarithms To condense a logarithm, we write the sum or difference of two or more logarithms as single expression. NOTE: You will be using properties of logarithms to do so.

14. Honors Algebra 214 Properties for Condensing Logarithmic Expressions (Working Backwards) Properties of Logarithms Let M, N, and b be any positive numbers, such that b ≠ 1. Product Rule: Quotient Rule: Power Rule: log b M+ log b N = log b (M ∙ N) log b M - log b N = log b (M /N) x log b M = log b M x

15. Honors Algebra 215 Example Condensing Logarithmic Expressions Write as a single logarithm: a.log 4 2 + log 4 32 = = a.log (4x - 3) – log x = log 4 64 3 log (4x – 3) x

16. Mrs. McConaughyHonors Algebra 216 NOTE: Coefficients of logarithms must be 1 before you condense them using the product and quotient rules. Write as a single logarithm: a.½ log x + 4 log (x-1) b.3 log (x + 7) – log x c. 2 log x + log (x + 1) = log x ½ + log (x-1) 4 = log √x (x-1) 4 = log (x + 7) 3 – log x = log (x + 7) 3 x = log x 2 + log (x + 1) = log x 2 (x + 1)

17. Honors Algebra 217 Check Point: Simplifying (Condensing) Logarithms a.log 3 20 - log 3 4 = b. 3 log 2 x + log 2 y = c. 3log 2 + log 4 – log 16 = log 3 (20/4) = log 3 5 log 2 x 3 y log 2 3 + log 4 – log 16 = log 32/16 =log 2

18. Honors Algebra 218 Example 1 Identifying the Properties of Logarithms State the property or properties used to rewrite each expression: Property:____________________________ log 2 8 - log 2 4 = log 2 8/4 = log 2 2 = 1 Property:____________________________ log b x 3 y = log b x 3 + log b 7 = 3log b x + log b 7 Property:____________________________ log 5 2 + log 5 6 = log 5 12 Quotient Rule (Property) Product Rule/Power Rule Product Rule (Property) Sometimes, it is necessary to use more than one property of logs when you expand/condense an expression.

19. Honors Algebra 219 Example Demonstrating Properties of Logs Use log 10 2 ≈ 0.031 and log 10 3 ≈ 0.477 to approximate the following: a. log 10 2/3 b. log 10 6 c. log 10 9 log 10 2 – log 10 3 0.031 – 0.477 – 0.466

20. 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!

21. Examples Find the value of log 2 37 Change to base 10 and use your calculator. log 37/log 2 Now use your calculator and round to hundredths. = 5.21 Log 7 99 = ? Change to base 10. Try it and see. log 3 81 log 4 256 log 2 1024

22. Let’s try some Working backwards now: write the following as a single logarithm.

23. Let’s try something more complicated... Condense the logs log 5 + log x – log 3 + 4log 5

24. Let’s try something more complicated... Expand