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171S5.4q Properties of Logarithmic Functions April 16, 2013 Logarithms of Products MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions and Graphs 5.4 Properties of Logarithmic Functions 5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest 5.4 Properties of Logarithmic Functions • Convert from logarithms of products, powers, and quotients to expressions in terms of individual logarithms, and conversely. The Product Rule For any positive numbers M and N and any logarithmic base a, loga MN = loga M + loga N. (The logarithm of a product is the sum of the logarithms of the factors.) Example Express as a single logarithm: Solution: Logarithms of Powers The Power Rule For any positive number M, any logarithmic base a, and any real number p, (The logarithm of a power of M is the exponent times the logarithm of M.) • Simplify expressions of the type logaax and . Nov 151:50 PM Nov 151:50 PM Logarithms of Quotients Examples The Quotient Rule For any positive numbers M and N, and any logarithmic base a, Express as a product. (The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.) Example Express as a difference of logarithms: Solution: Example Express as a single logarithm: Solution: Nov 151:50 PM Applying the Properties Examples Express each of the following in terms of sums and differences of logarithms. Nov 151:50 PM Example (continued) Solution: Solution: Example Express as a single logarithm: Solution: Nov 151:50 PM Nov 151:50 PM 1 171S5.4q Properties of Logarithmic Functions Examples Given that loga 2 ≈ 0.301 and loga 3 ≈ 0.477, find each of the following, if possible. April 16, 2013 Expressions of the Type loga ax The Logarithm of a Base to a Power For any base a and any real number x, loga ax = x. Solution: (The logarithm, base a, of a to a power is the power.) Examples Simplify. a) loga a8 b) ln et c) log 103k Solution: Cannot be found using these properties and the given information. Nov 151:50 PM Expressions of the Type a. loga a8 Nov 151:50 PM 441/2. Express as the sum of logarithms: log2 (8 . 64) A Base to a Logarithmic Power For any base a and any positive real number x, (The number a raised to the power loga x is x.) 441/4. Express as the sum of logarithms: log4 (64 . 4) Examples Simplify. 441/6. Express as the sum of logarithms: log 0.2x Solution: 441/8. Express as the sum of logarithms: ln ab Nov 151:50 PM 441/10. Express as a product: loga x 4 Nov 158:13 PM 441/18. Express as a difference of logarithms: loga (76 / 13) 441/12. Express as a product: ln y 5 441/20. Express as a difference of logarithms: ln (a / b) 441/14. Express as a product: logb Q 8 441/21. Express as a difference of logarithms: ln (r / s) 441/16. Express as a product: ln √a Nov 158:13 PM 441/22. Express as a difference of logarithms: logb (3 / w) Nov 158:13 PM 2 171S5.4q Properties of Logarithmic Functions 441/24. Express in terms of sums and differences of logarithms: 441/26. Express in terms of sums and differences of logarithms: Nov 158:13 PM 441/38. Express as a single logarithm and, if possible, simplify: ln 54 ln 6 441/42. Express as a single logarithm and, if possible, simplify: (2/5) loga x (1/3) loga y April 16, 2013 441/32. Express in terms of sums and differences of logarithms: 441/34. Express in terms of sums and differences of logarithms: Nov 158:13 PM 442/48. Express as a single logarithm and, if possible, simplify: 442/50. Express as a single logarithm and, if possible, simplify: (2 / 3) [ln (x2 9) ln (x + 3)] + ln (x + y) Nov 158:13 PM Nov 158:13 PM 442/54. Given that loga 2 ≈ 0.301, loga 7 ≈ 442/57. Given that loga 2 ≈ 0.301, loga 7 ≈ 0.845, and loga 11 ≈ 1.041, find each of the following, if possible. Round to nearest thousandth: loga 14 0.845, and loga 11 ≈ 1.041, find each of the following, if possible. Round to nearest thousandth: 442/56. Given that loga 2 ≈ 0.301, loga 7 ≈ 0.845, and loga 11 ≈ 1.041, find each of the following, if possible. Round to nearest thousandth: loga (1 / 7) Nov 158:13 PM 442/58. Given that loga 2 ≈ 0.301, loga 7 ≈ 0.845, and loga 11 ≈ 1.041, find each of the following, if possible. Round to nearest thousandth: loga 9 Nov 158:13 PM 3 171S5.4q Properties of Logarithmic Functions April 16, 2013 442/68. Simplify: logq q(√ 3) 442/70. Simplify: Nov 158:13 PM Apr 168:41 AM 442/72. Simplify: 442/74. Simplify: log 10 k Nov 158:13 PM Apr 164:12 PM 4