Summary of Exponential and Logarithmic Functions

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Summary of Exponential and Logarithmic Functions
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An Exponential Function is a function with
a ________ in the __________ .
variable
exponent
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Exponential Functions
Range
y
All Positive Real Numbers
x
Domain
All Real Numbers
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Exponential Decay
y
Exponential Growth Curve reflected across the
y-axis.
x
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Growth Rate Formula
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A Logarithm is an __________ .
exponent
A logarithm is the exponent to which the given
base is raised to get the given number.
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Exponential Form toLogarithmic Form
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Logarithmic Form to Exponential Form
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Logarithmic Functions
y
Range
All Real Numbers
x
Domain
All Positive Real Numbers
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Logarithm Properites
Product Property
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Logarithm Properites
Quotient Property
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Logarithm Properites
Power Property
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.SO, Now we can solve equations with a variable
in the exponent.
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How long will it take a population of 100 people
to grow to 150 people with a 3 growth rate?
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How long100 ? 150 _at_ 3 growth rate?
Now what?
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How long100 ? 150 _at_ 3 growth rate?
Logarithms to the Rescue!
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Solving an Exponential Equation

• Divide FIRST! Simplify
  so the base raised to a variable some number.
• Take the common log of both sides.
• Apply the power property to get the variable down
  out of the exponent.
• Then divide to solve for the variable.

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Solving Logarithmic Equations

• Use logarithm properties to simplify to
  get a single logarithm.
• (one side or both sides)
• If log A log B, then A B
  (Remove the logarithm from each side.)
• If log A constant, then change from logarithmic
  form to exponential form and then solve.

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Solving Logarithmic Equations
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Solving Logarithmic Equations
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Solving Logarithmic Equations
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Orders of Magnitude see p. 323

• The common logarithm of a positive quantity is
  its order of magnitude.
• Earthquakes Richter Scale
• Sound Intensity Decibels
• Ph scale for Chemical Acidity

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Newtons Law of Cooling

• The object of a heated object will cool to the
  temperature of the medium into which it is
  placed, such as the air or water.
• T Temperature
• T0 Initial Temperature
• Tm Temperature of medium
• t time
• k decay constant

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Newtons Law of Cooling
A cup of coffee is poured in a cup at a
temperature of 150º. Room temperature is
72º. Five minutes later the temperature of the
coffee is 100º. When will the coffee cool to 80º
?