Logarithmic Functions

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Logarithmic Functions Graphs, Lesson 3.2, page
388

• Objective To graph logarithmic functions, to
  convert between exponential and logarithmic
  equations, and find common and natural logarithms
  using a calculator.

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DEFINITION

• Logarithmic function inverse of exponential
  function
• If y bx, then the inverse is x by
• So y is the power which we raise b to in order to
  get x.
• Since we cant solve this for y, we change it to
  logarithmic form which is
• y logbx

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Think of logs like this

• logbN P and bp N
• Key b base, N number, P power
• Restrictions
• b gt 0 and b cannot equal 1
• N gt 0 because the log of zero or a negative
  number is undefined.

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Changing Exponential ? Log

• Log form gt logb N P
• Ex) log28 3
• Think A logarithm equals an exponent!
• Exponential form gt bP N
• Ex) 23 8

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Examples of Conversion

• Log Form logbN P Exponential Form bP N
• Log264 6
• Log101000 3
• Log416 2
• 25 32
• 104 10000
• 44 256

6
Rewrite the following exponential expression as a
logarithmic one.
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See Example 1, page 389.Check Point 1.

• Write each equation in its equivalent exponential
  form
• A) 3 log7x B) 2 logb25
• C) log426 y

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See Example 2, page 389.Check Point 2.

• Write each equation in its equivalent logarithmic
  form
• A) 25 x B) b3 27
• C) e y 33

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See Example 3, page 389.Check Point 3.

• Evaluate
• A) log10 100 B) log3 3
• C) log36 6

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See page 390.

• BASIC LOG PROPERTIES
• logb b 1
• logb 1 0
• INVERSE PROPERTIES OF LOGS
• logb bx x
• blogbx x

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Examples

• Check Point 4.
• A) log99 b) log8 1
• Check Point 5
• A) log7 78 b) 3log317

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Graphs

• Since exponential and logarithmic functions are
  inverses of each other, their graphs are also
  inverses.

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• Logarithmic function and exponential function are
  inverses of each other.
• The domain of the exponential function is all
  reals, so thats the domain of the logarithmic
  function.
• The range of the exponential function is xgt0, so
  the range of the logarithmic function is ygt0.

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See Example 6, page 391.

• Check Point 6
• Graph f(x) 3x and g(x) log3 x in the same
  rectangular coordinate system.

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Graph f(x) 3x.
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Now lets add f(x) log3x.(Simply find the
inverse of each point from f(x) 3x.)
(0, 1)
(1, 3)
(2, 9)
(3, 27)
(?1, 1/3)
(?2, 1/9)
(?3,1/27)
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See Characteristics of Graphs of Logs on page 392.

• See Table 3.4 on Transformations.

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Graphing Summary

• Logarithmic functions are inverses of exponential
  functions. Easier if rewrite as an exponential
  before graphing.
• 1. Choose values for y.
• 2. Compute values for x.
• 3. Plot the points and connect them with a
  smooth curve.
• Note that the curve does not touch or cross
  the y-axis.

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Comparing Exponential and Logarithmic Functions
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Domain Restrictions for Logarithmic Functions

• Since a positive number raised to an exponent
  (pos. or neg.) always results in a positive
  value, you can ONLY take the logarithm of a
  POSITIVE NUMBER.
• Remember, the question is What POWER can I
  raise the base to, to get this value?
• DOMAIN RESTRICTION

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See Example 7, page 393.

• Check Point 7 Find the domain of f(x)log4
  (x-5).

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Common Logarithms -- Intro

• If no value is stated for the base, it is assumed
  to be base 10.
• log(1000) means, What power do I raise 10 to, to
  get 1000? The answer is 3.
• log(1/10) means, What power do I raise 10 to, to
  get 1/10? The answer is -1.

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COMMON LOGARITHMS

• A common logarithm is a log that uses 10 as its
  base.
• Log10 y is written simply as log y.
• Examples of common logs are
• Log 100, log 50, log 26.2, log (1/4)
• Log button on your calculator is the common log

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Find each of the following common logarithms on a
calculator.

• Round to four decimal places.
• a) log 723,456
• b) log 0.0000245
• c) log (?4)

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Find each of the following common logarithms on a
calculator.
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Natural Logarithms -- Intro

• ln(x) represents the natural log of x, which has
  a basee
• What is e? If you plug large values into
  you get closer and closer to e.
• logarithmic functions that involve base e are
  found throughout nature
• Calculators have a button ln which represents
  the natural log.

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Natural Logarithms

• Logarithms, base e, are called natural
  logarithms.
• The abbreviation ln is generally used for
  natural logarithms.
• Thus, ln x means loge x.

ln button on your calculator is the natural
log
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Find each of the following natural logarithms on
a calculator.

• Round to four decimal places.
• a) ln 723,456
• b) ln 0.0000245
• c) ln (?4)

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