171S5.4p Properties of Logarithmic Functions. November 20, CHAPTER 5: Exponential and Logarithmic Functions. Examples. Express as a product.
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1 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. Simplify expressions of the type log a a x and. Nov 20 2:00 PM Logarithms of Products The Product Rule For any positive numbers M and N and any logarithmic base a, log a MN = log a M + log a N. (The logarithm of a product is the sum of the logarithms of the factors.) 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.) Nov 20 8:04 AM s Express as a product. 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.) Express as a difference 1
2 Applying the Properties s Express each of the following in terms of sums and differences of logarithms. (continued) s Given that loga and loga , find each of the following, if possible. Expressions of the Type log a a x The Logarithm of a Base to a Power For any base a and any real number x, loga a x = x. (The logarithm, base a, of a to a power is the power.) s Simplify. a) loga a 8 b) ln e t c) log 10 3k Cannot be found using these properties and the given information. a. loga a 8 Expressions of the Type A Base to a Logarithmic Power 441/2. Express as the sum log2 (8. 64) For any base a and any positive real number x, (The number a raised to the power log a x is x.) s Simplify. 441/4. Express as the sum log4 (64. 4) 2
3 441/6. Express as the sum log 0.2x 441/14. Express as a product: log b Q 8 441/8. Express as the sum ln ab 441/10. Express as a product: loga x 4 441/16. Express as a product: ln a 441/12. Express as a product: ln y 5 441/18. Express as a difference log a (76 / 13) 441/21. Express as a difference ln (r / s) 441/22. Express as a difference log b (3 / w) 441/20. Express as a difference ln (a / b) 441/24. Express in terms of sums and differences 441/34. Express in terms of sums and differences 441/32. Express in terms of sums and differences 441/26. Express in terms of sums and differences 3
4 441/38. Express as a single logarithm and, if ln 54 ln 6 442/48. Express as a single logarithm and, if 441/42. Express as a single logarithm and, if (2/5) loga x (1/3) loga y 442/50. Express as a single logarithm and, if (2 / 3) [ln (x 2 9) ln (x + 3)] + ln (x + y) 442/54. Given that loga , loga 7 loga /57. Given that loga , loga 7 442/56. Given that loga , loga 7 loga (1 / 7) 442/58. Given that loga , loga 7 loga 9 442/68. Simplify: log q q ( 3) 442/70. Simplify: Nov 20 10:25 AM 4
5 442/72. Simplify: 442/74. Simplify: log 10 k 5
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Powers and roots 6.1. Previous learning. Objectives based on NC levels and (mainly level ) Lessons 1 Squares, cubes and roots.
N 6.1 Powers and roots Previous learning Before they start, pupils should be able to: use index notation and the index laws for positive integer powers understand and use the order of operations, including