# Properties of Logarithms

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06-Jan-2016Category

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### 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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