3.4 Properties of Logarithmic Functions

3.4 Properties of Logarithmic Functions

3.3 Properties of Logarithmic Functions Properties of Logarithms Let b, R, and S be positive real numbers with b 1, and c any real number. Product Rule: log b ( RS ) log b R log b S R Quotient Rule: log b log b R log b S S c Power Rule: log b R c log b R Expanding the Logarithm of a Product Assuming x and y are positive, use properties of logarithms to write log (8xy4) as a sum of

logarithms or multiples of logarithms. 4 log(8 xy ) 4 log 8 log x log y 3 4 log 2 log x log y 3log 2 log x 4 log y Expanding the Logarithm of a Quotient Assuming x is positive, use properties of logarithms to write ln( x 2 5 / x) as a sum or

difference of logarithms or multiples of logarithms. 2 x 5 ln x x ln 2 5 12

x ln x 5 2 12 ln x 1 2 ln x 5 ln x 2 Condensing a Logarithmic Expression Assuming x and y are positive, write ln x5 2 ln

(xy) as a single logarithm. 5 5 ln x 2 ln( xy ) ln x ln( xy ) 5 2 2 2 ln x ln( x y ) 5

x ln 2 2 x y 3 x ln 2 y Change of Base When working with a logarithmic expression with an undesirable base, it is possible to change the expression into a quotient of logarithms with a different base. Change-of-Base Formula for Logarithms: For positive real numbers a, b, and x with a 1 and b 1,

log a x log b x . log a b Change-of-Base Formula The following two forms are useful in evaluating logarithms and graphing logarithmic functions. log x ln x OR log b x log b x log b ln b

Evaluating Logarithms by Changing the Base a.) ln16 log 3 16 2.523... 2.52 ln 3 b.) 1 log10 log 6 10 1.285... 1.29 log 6 log 6

ln 2 ln 2 ln 2 1 12 2 ln 1 2 ln1 ln 2 ln 2 c.) log Graphs of Logarithmic Functions with Base b Using the change-of-base formula we can rewrite any logarithmic function g(x) = logbx as

ln x 1 g ( x) ln x ln b ln b Graphing Logarithmic Functions Describe how to transform the graph of f(x) = ln x into the graph of the given function. a.) g(x) = log5x The graph is obtained by vertically shrinking the graph by a factor of1/ ln 5 0.62 Graphing Logarithmic Functions

Describe how to transform the graph of f(x) = ln x into the graph of the given function. b.) h(x) = log1/4x The graph is obtained by, in any order, a reflection across the x-axis and a vertical shrink by a factor of . 1 / ln 4 0.72 Vocabulary Check pg 243 1-5 Exercises pg 243 1-81 every other odd, 84

Recently Viewed Presentations

  • Ephesians Babiyes Ephesians  Ephesians Imitators ~ mimts  Ephesians

    Ephesians Babiyes Ephesians Ephesians Imitators ~ mimts Ephesians

    Ephesians 5:1-21. Rom. 1:24 ~ Therefore God also gave them up to uncleanness, in the lusts of their hearts, to dishonor their bodies among themselves, Rom. 1:26 ~ For this reason God gave them up to vile passions. For even...
  • The Muscular System Manual: The Skeletal Muscles of the Human ...

    The Muscular System Manual: The Skeletal Muscles of the Human ...

    The medial rotator muscles are active during the stance phase of gait performing their reverse action of ipsilateral rotation of the pelvis at the hip joint. The hip joint lateral rotators are primarily active during the stance phase of gait....
  • 'Try something new to-day': the range of research methods ...

    'Try something new to-day': the range of research methods ...

    'Try something new to-day' Sociologists gravitate towards two research methods: surveys and qualitative interviews . There is nothing wrong with these methods, but many other methods are available, including some that may be just as appropriate to use for your...
  • The Nishan Sahib - Sikhi Resources

    The Nishan Sahib - Sikhi Resources

    Fare the ropes which hold the nishan sahib in place. There are four of them which means people from all directions can receive a message from Guru Granth Sahib jee. Part G. Gis called the Thara. If the flag of...
  • Asee 2016 - Sbl

    Asee 2016 - Sbl

    Critical/creative thinking website developed to support the Quality Enhancement Plan (QEP) for improving students thinking skills. In order to familiarize students with various critical/creative thinking strategies, a free, online training resource, qepCafe, customized for electronic/computer technology, is being used.
  • The PRIME Theory of motivation and its application to smoking ...

    The PRIME Theory of motivation and its application to smoking ...

    ational Centre for Smoking . Cessation . and Training. June 2013. Robert West. Essential features of a stop smoking service. It is for smokers who want help with stopping. ... The PRIME Theory of motivation and its application to smoking...
  • SIA305: Top 3 Security and Privacy Challenges in Identity ...

    SIA305: Top 3 Security and Privacy Challenges in Identity ...

    Underlying crypto. Based on the Brands protocols. 30+ papers (from '93 onward) Evolution of PKI. MIT Press book, foreword by Ron Rivest. Issuance uses a "restrictive blind signature"
  • P.E.E. - Language Arts with Ms. Rose

    P.E.E. - Language Arts with Ms. Rose

    Erin, Alvin, Priyanka Dust of snow, written by Robert Frost is a simple yet meaningful poem, which uses imagery to really help us imagine what it would be like in his situation. "The way a crow, shook down on me,...