Exponential, Logarithmic Functions

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Exponential, Logarithmic Functions

• and other special functional forms

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Aim How do we graph exponential and logarithmic
functions?

• Learning ObjectivesSWBAT
• Apply simple laws of exponents and logarithms to
  solving exponential and logarithmic equations
• Sketch the graph of any exponential or
  logarithmic function
• Solve application problems involving growth and
  decay

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DO NOW

• Solve 4x 8 using exponents
• Solve 4x 8 using logs
• Use your calculator to sketch the graph of y
  2x. Write down all of the characteristics of
  this graph you can see (you may also refer to the
  table of values if you need to)

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Overview

• Inverse Functions
• Exponential Functions
• Logarithmic Functions
• Applications

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Inverse Functions

• If f is a function with domain A and range B,
  then f-1 is its inverse function if and only if
• for every
• and for every

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Alternate Definition
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• In other words, an inverse function
  undoes/reverses what the original function does.

Algebraically, one computes the inverse function
of f by solving the equation for x, and then
exchanging y and x to get
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Existence of the Inverse

• Not all functions have inverse functions.
• If a function has an inverse it is said to be
  invertible.
• If f is a real-valued function, then for f to
  have a valid inverse, it must pass the horizontal
  line test, that is a horizontal line y k placed
  on the graph of f must pass through f exactly
  once for all real k.

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Properties of Inverse functions
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Exponential Functions
Laws of Exponents
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Properties of Exponential Functions
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y
y 1
x
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y
y 1
x
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Logarithms
Laws of Logarithms
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Common and Natural Logarithms
The Common Logarithm is the logarithm with base
10. Simply denoted as log
The Natural Logarithm is the logarithm with base
e 2.71828 (irrational number). Simply denoted
as ln
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Logarithmic Functions
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Exponential and Logarithmic Functions
y
1
x
1
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Applications

• Growth and Decay
• Learning Curves