Properties of Logarithms

Transcription

1 Properties of Logarithms Accelerated Pre-Calculus Mr. Niedert Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 1 / 14

2 Properties of Logarithms 1 Change-of-Base Formula Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 2 / 14

3 Properties of Logarithms 1 Change-of-Base Formula 2 Rewriting Logarithmic Expressions Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 2 / 14

4 Change-of-Base Formula Most calculators have only two types of logarithm keys one that evaluates common logarithms (base 10) or one that evaluates natural logarithms (base e). Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 3 / 14

5 Change-of-Base Formula Most calculators have only two types of logarithm keys one that evaluates common logarithms (base 10) or one that evaluates natural logarithms (base e). Some of the newer calculators do not require you to use the Change-of-Base Formula to evaluate logarithms with different bases, but if yours does, you will need to know how to utilize the Change-of-Base Formula. Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 3 / 14

6 Change-of-Base Formula Most calculators have only two types of logarithm keys one that evaluates common logarithms (base 10) or one that evaluates natural logarithms (base e). Some of the newer calculators do not require you to use the Change-of-Base Formula to evaluate logarithms with different bases, but if yours does, you will need to know how to utilize the Change-of-Base Formula. Change-of-Base Formula Let a, b, and x be positive real numbers such that a 1 and b 1. Then log a x can be converted to a different base as follows. Base b Base 10 Base e log a x = log b x log log b a a x = log x log log a a x = ln x ln a Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 3 / 14

7 Changing Bases Using Common Logarithms and Natural Logarithms Practice For each of the logarithms below, use the Change-of-Base Formula to evaluate the logarithm using common logarithms. Then repeat the process using natural logarithms. Round to three decimal places. a log 4 25 b log 2 12 Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 4 / 14

8 Properties of Logarithms We know that the logarithmic functions are the inverse functions of exponential functions. Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 5 / 14

9 Properties of Logarithms We know that the logarithmic functions are the inverse functions of exponential functions. It then makes since that the properties of logarithms are similar to the properties of exponents. Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 5 / 14

10 Properties of Logarithms We know that the logarithmic functions are the inverse functions of exponential functions. It then makes since that the properties of logarithms are similar to the properties of exponents. Properties of Logarithms Let a be a positive number such that a 1, and let n be a real number. If u and v are positive real numbers, the following properties are true. Logarithm with Base a Natural Logarithm Product Property log a (uv) = log a u + log a v ln uv = ln u + ln v u Quotient Property log a v = log a u log a v ln u = ln u ln v v Power Property log a u n = n log a u ln u n = n ln u Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 5 / 14

11 Using Properties of Logarithms Practice Write each logarithm in terms of ln 2 and ln 3. a ln 6 b ln 2 27 Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 6 / 14

12 Using Properties of Logarithms Practice Find the exact value of each expression without using a calculator. a log b ln e 6 ln e 2 Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 7 / 14

13 Properties of Logarithms (Part 1 of 2) Assignment Part 1: pg. 243 VC #1-5, Ex. #2-8 even, 9-22, even Part 2: pg Ex. #39-79 EOO, 80 Assignment: pg VC #1-5, Ex. #2-8 even, 9-22, even, EOO, 80 Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 8 / 14

14 Expanding Logarithmic Expressions Example Expand each logarithmic expression. a log 4 5x 3 y 3x 5 b ln 7 Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 9 / 14

15 Expanding Logarithmic Expressions Practice Expand each logarithmic expression. a log 3x 2 y 4x + 1 b ln 8 Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 10 / 14

16 Expanding vs. Condensing In the previous example and practice problems, the properties of logarithms were used to expand logarithmic expressions. Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 11 / 14

17 Expanding vs. Condensing In the previous example and practice problems, the properties of logarithms were used to expand logarithmic expressions. We can reverse this process, as we will in the following example and practice problems, and the properties of logarithms are used to condense logarithmic expressions. Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 11 / 14

18 Condensing Logarithmic Expressions Example Condense each logarithmic expression. a 1 2 log x + 3 log(x + 1) b 2 ln(x + 2) ln x c 1 3 [log 2 x + log 2 (x + 1)] Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 12 / 14

19 Condensing Logarithmic Expressions Practice Condense each logarithmic expression. a 1 3 log x + 5 log(x 3) b 4 ln(x 4) 2 ln x c 1 5 [log 3 x + log 3 (x 2)] Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 13 / 14

20 Properties of Logarithms (Part 2 of 2) Assignment Part 1: pg. 243 VC #1-5, Ex. #2-8 even, 9-22, even Part 2: pg Ex. #39-79 EOO, 80 Assignment: pg VC #1-5, Ex. #2-8 even, 9-22, even, EOO, 80 Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 14 / 14