unit 5: modeling with exponential & logarithmic functions ms. c. taylor
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Unit 5: Modeling with Exponential & Logarithmic FunctionsMs. C. Taylor
Warm-Up
Identify the value of b in the following:
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.
Warm-Up
Rewrite using exponent rules
Logarithms
Suppose b>0 and b≠1. For x>0, there is a number y such that if and only if
LogarithmicExponential Form
ExponentialLogarithmic Form
Inverse Property of Exponents & Logarithms
LogarithmicExponential Inequality
If
If
Property of Equality for Logarithmic Functions
If b is a positive number other than 1, then if and only if
Example: If , then
Property of Inequality for Logarithmic Functions
If , then if and only if, and if and only if
If , then
Product Property of Logarithms
For all positive numbers m, n, and b, where b≠1,
Example #1
Expand the following logarithms:
Example #2
Use to approximate the value of Use to approximate the value of
Quotient Property of Logarithms
For all positive numbers m, n, and b, where ,
Example #3
Expand the following logarithms:
Example #4
Use and to approximate Use and to approximate
Power Property of Logarithms
For any real number p and positive numbers m and b, where ,
Examples
Given , approximate the value of
Given , approximate the value of
Warm-Up
Expand the following:
Find Common Logarithms
Change of Base Formula
For all positive numbers, a, b, and n, where and ,
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.
Evaluate Natural Base Expressions
Evaluate Natural Logarithmic Expressions
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.
Warm-UpEvaluate the following
Warm-Up
Use the properties of logarithms to rewrite:
Inverse Property of Base e & Natural
Logarithms
Evaluate Logarithmic Expressions
Solve Logarithmic Equations
Solve Equations with Logarithms on Both
Sides
Solve
Solve Equations using Properties of Logarithms
Warm-Up
log 𝑥− log (𝑥−1 )=log (3 𝑥+12)
Solve Exponential Equations using
Logarithms
Solve Base e Equations
Solve Natural Log Equations & Inequalities