unit 5: modeling with exponential & logarithmic functions ms. c. taylor
TRANSCRIPT
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Unit 5: Modeling with Exponential & Logarithmic FunctionsMs. C. Taylor
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Warm-Up
Identify the value of b in the following:
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Graphing Exponential Equations
The graph will approach the axis but will never touch.
Asymptote for the function will approach the x-axis.
Asymptote for the inverse function will approach the y-axis.
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Warm-Up
Rewrite using exponent rules
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Logarithms
Suppose b>0 and b≠1. For x>0, there is a number y such that if and only if
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LogarithmicExponential Form
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ExponentialLogarithmic Form
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Inverse Property of Exponents & Logarithms
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LogarithmicExponential Inequality
If
If
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Property of Equality for Logarithmic Functions
If b is a positive number other than 1, then if and only if
Example: If , then
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Property of Inequality for Logarithmic Functions
If , then if and only if, and if and only if
If , then
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Product Property of Logarithms
For all positive numbers m, n, and b, where b≠1,
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Example #1
Expand the following logarithms:
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Example #2
Use to approximate the value of Use to approximate the value of
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Quotient Property of Logarithms
For all positive numbers m, n, and b, where ,
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Example #3
Expand the following logarithms:
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Example #4
Use and to approximate Use and to approximate
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Power Property of Logarithms
For any real number p and positive numbers m and b, where ,
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Examples
Given , approximate the value of
Given , approximate the value of
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Warm-Up
Expand the following:
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Find Common Logarithms
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Change of Base Formula
For all positive numbers, a, b, and n, where and ,
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ExamplesExpress in terms of common
logarithms. Then approximate its value to four decimal places.
Express in terms of common logarithms. Then approximate its value to four decimal places.
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Evaluate Natural Base Expressions
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Evaluate Natural Logarithmic Expressions
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Equivalent Expressions
If something has an e in it then that will become a ln.
If something has an ln in it then it will become e raised to a power.
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Warm-UpEvaluate the following
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Warm-Up
Use the properties of logarithms to rewrite:
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Inverse Property of Base e & Natural
Logarithms
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Evaluate Logarithmic Expressions
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Solve Logarithmic Equations
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Solve Equations with Logarithms on Both
Sides
Solve
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Solve Equations using Properties of Logarithms
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Warm-Up
log 𝑥− log (𝑥−1 )=log (3 𝑥+12)
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Solve Exponential Equations using
Logarithms
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Solve Base e Equations
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Solve Natural Log Equations & Inequalities