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171S5.4 Properties of Logarithmic Functions 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 Logarithms of Products 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.) November 21, 2011 5.4 Properties of Logarithmic Functions • Convert from logarithms of products, powers, and quotients to expressions in terms of individual logarithms, and conversely. • Simplify expressions of the type logaax and . 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.) Example Express as a single logarithm: Solution: 1 171S5.4 Properties of Logarithmic Functions Examples Express as a product. November 21, 2011 Logarithms of Quotients The Quotient Rule For any positive numbers M and N, and any logarithmic base a, (The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.) Example Express as a difference of logarithms: Applying the Properties Examples Express each of the following in terms of sums and differences of logarithms. Solution: Example Express as a single logarithm: Solution: 2 171S5.4 Properties of Logarithmic Functions Example (continued) Solution: November 21, 2011 Example (continued) Solution: Example Examples Given that loga 2 ≈ 0.301 and loga 3 ≈ 0.477, find each of the following, if possible. Express as a single logarithm: Solutio Solution: 3 171S5.4 Properties of Logarithmic Functions Examples (continued) loga 2 ≈ 0.301 and loga 3 ≈ 0.477 November 21, 2011 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. Cannot be found using these properties and the given information. (The logarithm, base a, of a to a power is the power.) Examples Expressions of the Type Simplify. a) loga a8 b) ln et c) log 103k A Base to a Logarithmic Power For any base a and any positive real number x, Solution: a. loga a8 (The number a raised to the power loga x is x.) 4 171S5.4 Properties of Logarithmic Functions Examples November 21, 2011 433/2. Express as the sum of logarithms: log2 (8 . 64) Simplify. Solution: 433/14. Express as a product: logb Q 8 433/18. Express as a difference of logarithms: loga (76 / 13) 5 171S5.4 Properties of Logarithmic Functions November 21, 2011 433/32. Express in terms of sums and differences of logarithms: logc ∛(y3 z / x4) 433/24. Express in terms of sums and differences of logarithms: loga x3 y2 z 433/38. Express as a single logarithm and, if possible, simplify: ln 54 ln 6 433/42. Express as a single logarithm and, if possible, simplify: (2/5) loga x (1/3) loga y 6 171S5.4 Properties of Logarithmic Functions November 21, 2011 433/56. 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) 433/50. Express as a single logarithm and, if possible, simplify: (2 / 3) [ln (x2 9) ln (x + 3)] + ln (x + y) 433/68. Simplify: logq q(√ 3) 433/72. Simplify: eln x3 7