Properties of Logarithms

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Properties of Logarithms. They’re in Section 3.4a. Proof of a Prop ‘o Logs. Let. and. In exponential form:. Let’s start with the product of R and S :. A Prop ‘o Logs!!!. Properties of Logarithms. Let b , R , and S be positive real numbers with b = 1, and c any real number. - PowerPoint PPT Presentation

Transcript of Properties of Logarithms

  • Properties of LogarithmsTheyre in Section 3.4a

  • Proof of a Prop o LogsLetandIn exponential form:Lets start with the product of R and S:A Prop o Logs!!!

  • Properties of LogarithmsLet b, R, and S be positive real numbers with b = 1, and cany real number. Product Rule: Quotient Rule: Power Rule:

  • Guided PracticeAssuming x and y are positive, use properties of logarithmsto write the given expression as a sum of logarithms ormultiples of logarithms.

  • Guided PracticeAssuming x is positive, use properties of logarithms to writethe given expression as a sum or difference of logarithmsor multiples of logarithms.

  • Guided PracticeAssuming x and y are positive, use properties of logarithmsto write the given expression as a single logarithm.

  • Of the eight relationships suggested here, four are true and fourare false (using values of x within the domains of both sides ofthe equations). Thinking about the properties of logarithms,make a prediction about the truth of each statement. Then testeach with some specific numerical values for x. Finally, comparethe graphs of the two sides of the equation.1.3.5.7.2.4.6.8.These four statements are TRUE!!!

  • A few more problemsAssuming x and y are positive, use properties of logarithms towrite the expression as a sum or difference of logarithms ormultiples of logarithms.

  • And still a few more problemsAssuming x, y, and z are positive, use properties of logarithms towrite the expression as a single logarithm.

  • WhiteboardWrite as a single logarithmic expression:

  • Lets do an explorationHow do we evaluate ?First, switch to exponential form.Apply ln to both sides.Use the power rule.Set equal to y:Divide by ln4.We just provedthe C.O.B.!!!

  • Change-of-Base Formula for LogarithmsFor positive real numbers a, b, and x with a = 1and b = 1,Because of our calculators, the two most common forms:

  • Guided PracticeEvaluate each of the following.1.2.3.

  • Guided PracticeWrite the given expression using only natural logarithms.1.2.

  • Guided PracticeWrite the given expression using only common logarithms.1.2.

  • Graphs of Logarithmic Functions with Base bRewrite the given function using the change-of-base formula. Every logarithmic function is a constant multiple of the natural logarithmic function!!!If b > 1, the graph of g(x) is a vertical stretch or shrink of thegraph of the natural log function by a factor of 1/(ln b).If 0 < b < 1, a reflection across the x-axis is required as well.

  • More Guided PracticeDescribe how to transform the graph of the natural logarithmfunction into the graph of the given function. Sketch the graphby hand and support your answer with a grapher.1.Vertical shrink by a factorof approximately 0.621.How does the graph look???

  • More Guided PracticeDescribe how to transform the graph of the natural logarithmfunction into the graph of the given function. Sketch the graphby hand and support your answer with a grapher.2.Reflect across x-axis, Verticalshrink by a factor of 0.721How does the graph look???

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