1. Properties of Logarithms

2. Product property of logarithms For all positive numbers m, n, and b, where b  1, log b mn = log b m + log b n.

3. Example1. Given log 3 5  1.4650, find each logarithm. a. log 3 45 log 3 (95) log 3 9 + log 3 5 2 + 1.4650 = 3.4650

4. Example1. Given log 3 5  1.4650, find each logarithm. b. log 3 25 log 3 (55) log 3 5 + log 3 5 1.4650 + 1.4650 = 2.9300

5. Quotient property of logarithms For all positive numbers m, n, and b, where b  1, log b m/n = log b m - log b n.

6. Example2. Given log 4 5  1.1610 and log 4 15  1.9534, find each logarithm. a. log 4 5/16 log 4 5 - log 4 16 1.1610 - 2 = -0.8390

7. Example2. Given log 4 5  1.1610 and log 4 15  1.9534, find each logarithm. b. log 4 3 How can I rewrite 3 using 5 and 15? log 4 15/5 log 4 15 - log 4 5 1.9534 - 1.1610 = 0.7924

8. Example3. The pH of a substance is the concentration of hydrogen ions, [H+], measured in moles of hydrogen per liter of substance. It is given by the formula, pH = log 10 (1/[H+]) Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2.

9. Example3. The pH of acid rain. It is given by the formula, pH = log 10 (1/[H+]) Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2. 4.2 = log 10 (1/[H+]) 4.2 = log 10 1 - log 10 [H+]

10. Example3. The pH of acid rain. Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2. 4.2 = log 10 (1/[H+]) 4.2 = log 10 1 - log 10 [H+] 4.2 = 0 - log 10 [H+] 4.2 = -log 10 [H+]

11. Example3. The pH of acid rain. 4.2 = log 10 1 - log 10 [H+] 4.2 = 0 - log 10 [H+] 4.2 = -log 10 [H+] -4.2 = log 10 [H+] 10 -4.2 = [H+] [H+] = 10 -4.2  0.000063

12. Power property of logarithms For any real number p and positive numbers m, and b, where b  1, log b m p = plog b m.

13. Example4. Solve a. 2log 3 6 - (1/4)log 3 16 = log 3 x b. log 10 z + log 10 (z+3) = 1

14. Example4. Solve a. 2log 3 6 - (1/4)log 3 16 = log 3 x log 3 6 2 - log 3 16 1/4 = log 3 x log 3 36 - log 3 2 = log 3 x log 3 36/2 = log 3 x log 3 18 = log 3 x x = 18

15. Example4. Solve b. log 10 z + log 10 (z+3) = 1 log 10 z(z+3) = 1 z(z+3) = 10 1 z 2 + 3z - 10 = 0 (z+5)(z-2) = 0 z = -5 or z = 2 z = 2 is the solution.

16. Example5. Rewrite as one logarithm log 10 2 + log 10 (x+9) + log 10 (y+6) The properties allow us to rewrite these two additions as a single multiplication problem. log 10 [2(x+9)(y+6)] log 10 (2xy+12x+18y+108)

17. Example6. Rewrite as an equivalent logarithmic expression log a √ 47 69 log a 47 69 1212 log a 47 69 1212 log a 47-log a 69 1212 1212 1212

18. Example7. Rewrite as an equivalent logarithmic expression log a x 4 y 9 z 5 1616 log a x 4 y 9 z 5 1616 log a x 4 y 9 -log a z 5 1616 log a √ x 4 y 9 z 5 6 log a x 4 + log a y 9 -log a z 5 1616

19. Example7. Rewrite as an equivalent logarithmic expression log a x 4 + log a y 9 -log a z 5 1616 4log a x + 9log a y-5log a z 1616 log a x + log a y- log a z 2323 3232 5656