1. Evaluation Logarithms Common Log log b 1 = 0log 1 = 0 log b b = 1log 10 = 1 log b b m = mlog 10 s = s b log b x = x 10 log x = x log b 0 is not defined log b (-x) is not defined For x > 0 and b  1, 8.3 Laws of Logarithms log 4 4 = 1 log 8 1 = 0 3 log 3 6 = 6 log 5 5 3 = 3 2 log 2 7 = 7 Examples Log a x = Log b x Log b a Log 2 x = Log x Log 2 Math 30-11

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3. Addition Law of Logarithms Let p = log a x and q = log a y Convert x = a p and y = a q xy = a p+q Convert p + q = log a (xy) p + q = log a x + log a y log a (xy) = log a x + log a y Math 30-13 log a (xy) = log a x + log a y

4. Math 30-14 Express as the sum of two logarithms.

5. Math 30-15

6. Subtraction Law of Logarithms Let p = log a x and q = log a y Convert x = a p and y = a q x/y = a p-q Convert p - q = log a ( x / y ) p - q = log a x - log a y log a ( x / y ) = log a x - log a y Math 30-16 log a ( x / y ) = log a x - log a y

7. Math 30-17 Express as the difference of two logarithms.

8. Comparing Exponent Laws to Laws of Logarithms log a ( x / y ) = log a x - log a y log a (xy) = log a x + log a y a m.a n = a m+n a m /a n = a m-n Express as a sum and difference of logarithms = log 3 27 + log 3 3 - log 3 9 = 3 + 1 - 2 = 2 Math 30-18

9. Evaluate: log 2 10 + log 2 12.8 = log 2 (10 x 12.8) = log 2 (128) = log 2 (2 7 ) = 7 Applying Laws of Logarithms Express as a single log: log 3 A - log 3 B - log 3 C log 3 A - log 3 B + log 3 C Simplify: log 5 50 - log 5 10 log 5 5 = 1 Math 30-19

10. Given log 7 9 = a, determine an expression in terms of a for log 7 63. log 7 63 = log 7 (9 x 7) = log 7 9 + log 7 7 = a + 1 Simplify: log 4 5a + log 4 8a 3 - log 4 10a 4 log 4 4 = 1 Simplifying Logarithms Math 30-110 If and, express each of the following in terms of x and y

11. Math 30-111

12. Power Law: log b m n = log b m a log 2 4 1/3 3log 2 5 n log b m Math 30-112 log b (x n ) = nlog b x

13. Applying the Power Laws Given that log 3 a = 6 and log 3 b = 5 determine the value of log 3 (9ab 2 ), where a, b > 0 Write as a single logarithm, where x, y, z > 0 Math 30-113

14. Given log 6 2 = a and log 6 5 = b rewrite in terms of a and b. Math 30-114

15. The expression is equivalent to Evaluate: a) = 4 b) = 2 Math 30-115

16. Assignment: State whether the following are True or False for logarithms to every base. a) log 2 + log 3 = log 5 False b) log 4 + log 3 = log 12 True c) log 10 + log 10 = log 100True d) log 2 x log 3 = log 6 False e) log 3 2 + log 3 -2 = 0 True f) False g) False, think of change of base. Math 30-116

17. Assignment State whether the following are True or False for logarithms to every base. a) log 5 -2 = -2log 5 True b) True c)False d)False e) True f )False Page 400 1a,c, 2, 3, 5, 6, 8, 9, 10, 11, 12, Math 30-117