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9-2 Algebra 2B 2017: Logarithms and Logarithmic Functions In general, the inverse of y = b x is x = b y. In x = b y, y is called the logarithm of x. It is usually written as y = log b x and is read y equals log base b of x. New Vocabulary logarithm logarithmic function logarithmic equation logarithmic inequality Write each equation in logarithmic form. 1. 53 125 2. 70 1 41 1 81 4. 34 5. 3 f(x) = b x 3. 34 81 1 64 g[f(x)] = x f (log b x) = x g(b ) = x x 6. 7776 6 b log b = x Write each equation in exponential form. 7. log6 216 3 1 81 8. log2 64 6 9. log3 4 1 2 10. log10 0.00001 5 3 5 11. log25 5 12. log32 8 Evaluate each expression. 13. log9 1 1 3 17. log3 1 49 14. log8 4 1 256 18. log4 15. log7 16. log6 64 19. log9 9(n 1) 20. 2log2 32 Solve each equation or inequality. Check your solutions. 21. log15 x 3 22. log7 q 0 24. logy 16 4 25. logn 3 1 8 g(x) = log b x f [g(x)] = x 1 5 inverses 23. log6 (2y 8) 2 26. logb 1024 5 x log b b x = x Proper ties of Logarithms Suppose m and n are positive numbers, b is a positive number other than 1, and p is any real number. Then the following properties hold. Property Definition Example Product logb mn logb m logb n log3 9x log3 9 log3 x Quotient m logb logb m logb n n log14 log1 4 log1 5 Power logb m p p logb m log2 8x x log2 8 Equality If logb m logb n, then m n. log8 (3x 4) log8 (5x 2) so, 3x 4 5x 2 4 5 4 4 27. log8 (3x 7) log8 (7x 4) 28. log7 (8x 20) log7 (x 6) 29. log3 (x2 2) log3 x 30. INVESTING Cody invests $1000 in a savings account that pays 4% interest compounded annually. The value of the account A at the end of five years can be determined from the equation log A log[1000(1 0.04)5]. Find the value of A to the nearest dollar. 31. FISH The population of silver carp has been growing in the Mississippi River. About every 3 years, the population doubles. Write logarithmic expression that gives the number of years it will take for the population to increase by a factor of ten. 33. LOGARITHMS Ariel knows that logb x 3 and logb y 5. She knows that this is the same as knowing that b3 x and b5 y. Multiply these two equations together and rewrite it as an equation involving logarithms. What is logb xy? 32. POWERS Jordan tries to solve the equation log4 2x 5. She got the wrong answer. What was her mistake? What should the correct answer be? 34. DIGITS Jennifer, a programmer, wants to write a formula that tells how many digits there are in a number n, where n is a positive integer. For example, if n 343, the formula should evaluate to 3 and if n 10,000, the formula should evaluate to 5. Suppose 8 log10 n 9. How many digits does n have? 1. log4 2x 5 2. 2x 45 3. x 25 4. x 32