Properties of Logarithms

Transcription

1 Properties of Logarithms MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011

2 Objectives In this lesson we will learn to: use the change-of-base formula to rewrite and evaluate logarithmic expressions, use properties of logarithms to evaluate or rewrite logarithmic expressions, use properties of logarithms to expand or condense logarithmic expressions, use logarithmic functions to model and solve real-world problems.

3 Change of Base Most calculators include keys to evaluate common (base 10) and natural (base e) logarithms. Question: how can we evaluate logarithms with general base a (0 < a and a 1)?

4 Change of Base Most calculators include keys to evaluate common (base 10) and natural (base e) logarithms. Question: how can we evaluate logarithms with general base a (0 < a and a 1)? 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

5 Change of Base Most calculators include keys to evaluate common (base 10) and natural (base e) logarithms. Question: how can we evaluate logarithms with general base a (0 < a and a 1)? 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 Remark: logarithms with base a are constant multiples of 1 logarithms with base b. The constant multiple is log b a.

6 Examples Use a calculator to evaluate the following logarithmic expressions. log 7 4 = log 1/4 5 = log = log =

7 Examples Use a calculator to evaluate the following logarithmic expressions. log 7 4 = log 1/4 5 = log = log =

8 Examples Use a calculator to evaluate the following logarithmic expressions. log 7 4 = log 1/4 5 = log = log =

9 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. Product Property: log a (u v) = log a u + log a v Quotient Property: log a u v = log a u log a v Power Property: log a u n = n log a u Remark: in particular these properties hold for common and natural logarithms.

10 Examples Rewrite, simplify, and if possible evaluate each of the following logarithmic expressions without a calculator. ln 6 e 2 log

11 Examples Rewrite, simplify, and if possible evaluate each of the following logarithmic expressions without a calculator. ln 6 e 2 ln 6 e 2 = ln 6 ln e2 = ln 6 2 ln e = ln 6 2 log

12 Examples Rewrite, simplify, and if possible evaluate each of the following logarithmic expressions without a calculator. ln 6 e 2 ln 6 e 2 = ln 6 ln e2 = ln 6 2 ln e = ln 6 2 log log = log 3 (3 4 ) 3 = log = 12 log 3 3 = 12

13 Expanding Logarithmic Expressions Use the product, quotient, and power properties of the logarithm as appropriate to expand each of the following logarithmic expressions. log xy 4 z 5 ln x 2 (x + 2)

14 Expanding Logarithmic Expressions Use the product, quotient, and power properties of the logarithm as appropriate to expand each of the following logarithmic expressions. log xy 4 z 5 ln x 2 (x + 2) log xy 4 z 5 = log x + 4 log y 5 log z

15 Expanding Logarithmic Expressions Use the product, quotient, and power properties of the logarithm as appropriate to expand each of the following logarithmic expressions. log xy 4 z 5 ln x 2 (x + 2) log xy 4 z 5 = log x + 4 log y 5 log z ln x 2 (x + 2) = 1 2 (ln x 2 + ln(x + 2)) = ln x + 1 ln(x + 2) 2 if x > 0.

16 Condensing Logarithmic Expressions Use the product, quotient, and power properties of the logarithm as appropriate to condense each of the following logarithmic expressions. 3 log 3 x + 4 log 3 y 4 log 3 z 4[ln z + ln(z + 3) 2 ln(z 3)]

17 Condensing Logarithmic Expressions Use the product, quotient, and power properties of the logarithm as appropriate to condense each of the following logarithmic expressions. 3 log 3 x + 4 log 3 y 4 log 3 z 3 log 3 x + 4 log 3 y 4 log 3 z = log 3 x 3 + log 3 y 4 log 3 z 4 = log 3 x 3 y 4 z 4 4[ln z + ln(z + 3) 2 ln(z 3)]

18 Condensing Logarithmic Expressions Use the product, quotient, and power properties of the logarithm as appropriate to condense each of the following logarithmic expressions. 3 log 3 x + 4 log 3 y 4 log 3 z 3 log 3 x + 4 log 3 y 4 log 3 z = log 3 x 3 + log 3 y 4 log 3 z 4 = log 3 x 3 y 4 z 4 4[ln z + ln(z + 3) 2 ln(z 3)] [ 4[ln z+ln(z+3) 2 ln(z 3)] = 4 ln ] z(z + 3) (z 3) 2 = ln [ ] z(z + 3) 4 (z 3) 2

19 Application The relationship between the number of decibels β and the intensity of a sound I in watts per square meter is given be ( ) I β = 10 log Find the difference in loudness between an average office with an intensity of watts per square meter and a broadcast studio with an intensity of watts per square meter.

20 Application The relationship between the number of decibels β and the intensity of a sound I in watts per square meter is given be ( ) I β = 10 log Find the difference in loudness between an average office with an intensity of watts per square meter and a broadcast studio with an intensity of watts per square meter. ( ) For the office, β = 10 log 51 db ( ) For the studio, β = 10 log 25 db The difference is approximately 26 db.

21 Homework Read Section 3.3. Exercises: 1, 5, 9, 13,..., 85, 89