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Logarithmic Functions. Lesson 2.5. How to Graph These Numbers?. Consider the vast range of the numbers. How to Graph These Numbers?. What's wrong with this picture?. How to Graph These Numbers?. What's wrong with this picture?. We need a way to set a scale that fits all the data.

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how to graph these numbers
How to Graph These Numbers?
  • Consider thevast rangeof the numbers
how to graph these numbers1
How to Graph These Numbers?
  • What's wrongwith thispicture?
how to graph these numbers2
How to Graph These Numbers?
  • What's wrongwith thispicture?

We need a way toset a scale that fits all the data

how to graph these numbers3
How to Graph These Numbers?
  • The solution:Set the scaleto be the exponentof the distance

This is called a logarithmic scale

a new function
A New Function
  • Consider the exponential function y = 10x
  • Based on that function, declare a new function x = log10y
  • You should be able to see that these are inverse functions
  • In general
  • The log of a numberis an exponent
properties of logarithms
Properties of Logarithms
  • Note box on page 105 of text
  • Most used properties
change of base theorem
Change of Base Theorem
  • To find the log of a number for a base other than 10 or e …
    • Use
  • Where b can be any base
    • Typically 10 or e
    • Available on calculator

Note new spreadsheet assignment on Blackboard

change of base theorem1
Change of Base Theorem
  • Create a function for your calculator
    • Definefunction
    • Try it
    • Verify
solving log equations
Solving Log Equations
  • Use definition of logarithm
    • Rewrite log equation as an exponential equation
  • Result x = 32
solving exponential equations
Solving Exponential Equations
  • Use property of logarithms
  • Consider
    • Isolate exponential expression
    • Take ln of both sides
    • Solve for x
doubling time
Doubling Time
  • What if inflation is at the 5% rate …
  • How long until prices double?
  • Strategy
    • Divide through by P
    • Take log of both sides
    • Bring t out as coefficient
    • Solve for t
assignment
Assignment
  • Lesson 2.5
  • Page 121
  • Exercises 1 – 79 EOO