Properties of Logarithms

1. 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 Honors Algebra 2

2. Part 1:Expanding Logarithms Honors Algebra 2

3. PROPERTY: The Product Rule (Property) 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. Honors Algebra 2

4. Example Expanding a Logarithmic Expression Using Product Rule log (4x) = log 4 + log x is The logarithm of a product The sum of the logarithms. Use the product rule to expand: • log4 ( 7 • 9) = _______________ • log ( 10x) = ________________ = ________________ log4 ( 7) + log 4(9) log ( 10) + log (x) 1 + log (x)

5. 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. Honors Algebra 2

6. ExampleExpanding a Logarithmic Expression Using Quotient Rule log (x/2) = log x - log 2 is The logarithm of a quotient The difference of the logarithms. Use the quotient rule to expand: • log7 ( 14 /x) = ______________ • log ( 100/x) = ______________ = ______________ log7 ( 14) - log 7(x) log ( 100) - log (x) 2 - log (x)

7. PROPERTY: The Power Rule (Property) The Power Rule Let M, N, and b be any positive numbers, such that b ≠ 1. log b Mx = 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 Honors Algebra 2

8. ExampleExpanding a Logarithmic Expression Using Power Rule Use the power rule to expand: • log5 74= _______________ • log √x = ________________ = ________________ 4log5 7 log x 1/2 1/2log x Honors Algebra 2

9. Summary:Properties for Expanding Logarithmic Expressions log b (M ∙ N ) = log b M+ log b N log b (M / N ) = log b M - log b N log b Mx = x log b M NOTE: In all cases, M > 0 and N >0. Honors Algebra 2

10. Check Point: Expanding Logarithmic Expressions Use logarithmic properties to expand each expression: • logb x2√y b. log63√x 36y4 log b x2 + logb y1/2 log 6 x1/3 - log636y4 2log b x + ½ logb y log 6 x1/3 - (log636 + log6y4) 1/3log 6 x - log636 - 4log6y 2 Honors Algebra 2

11. Check Point: Expanding Logs NOTE: You are expanding, not condensing (simplifying) these logs. Expand: log 2 3xy2 log 8 26(xy)2 = log 2 3 + log 2 x + 2log 2y = log 8 26 + log 8 x2 + log 8y2 = 6log 8 2 + 2log 8 x + 2log 8y Honors Algebra 2

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

13. 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. Honors Algebra 2

14. Properties for Condensing Logarithmic Expressions (Working Backwards) 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 Mx Honors Algebra 2

15. Example Condensing Logarithmic Expressions Write as a single logarithm: • log4 2 + log 4 32 = = • log (4x - 3) – log x = log 4 64 3 log (4x – 3) x Honors Algebra 2

16. NOTE:Coefficients of logarithms must be 1 beforeyou condense them using the product and quotient rules. Write as a single logarithm: • ½ log x + 4 log (x-1) • 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 x2 + log (x + 1) = log x2 (x + 1) Honors Algebra 2

17. Check Point: Simplifying (Condensing) Logarithms • 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 3y log 23 + log 4 – log 16= log 32/16 =log 2 Honors Algebra 2

18. Example 1 Identifying the Properties of Logarithms Sometimes, it is necessary to use more than one property of logs when you expand/condense an expression. State the property or properties used to rewrite each expression: Quotient Rule (Property) Property:____________________________ log 2 8 - log 2 4 = log 2 8/4 = log 2 2 = 1 Property:____________________________ log b x3 y = log b x3 + log b 7 = 3log b x + log b 7 Property:____________________________ log 5 2 + log 5 6 = log 512 Product Rule/Power Rule Product Rule (Property) Honors Algebra 2

19. 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 log10 2 – log10 3 0.031 – 0.477 0.031 – 0.477 – 0.466 Honors Algebra 2

20. Change of Base Formula • Example log58 = • This is also how you graph in another base. Enter y1=log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10!

21. Examples • Find the value oflog2 37 • Change to base 10 and use your calculator. log 37/log 2 • Now use your calculator and round to hundredths. = 5.21 • Log7 99 = ? • Change to base 10.  Try it and see. • log3 81 • log4 256 • log2 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