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

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Presentation transcript:

Properties of Logarithms

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

Example1. Given log 3 5  , find each logarithm. a. log 3 45 log 3 (95) log log =

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

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.

Example2. Given log 4 5  and log 4 15  , find each logarithm. a. log 4 5/16 log log =

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

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.

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 = log 10 (1/[H+]) 4.2 = log log 10 [H+]

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

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

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.

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

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

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.

Example5. Rewrite as one logarithm log 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)

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

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

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