Logarithms%20and%20Logarithmic%20Functions

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Logarithms and Logarithmic Functions

• Coach Baughman
• November 20, 2003
• Algebra II

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Objectives

• The students will identify a logarithmic
  function. (Knowledge) (Mathematics, Algebra II,
  6.a)
• The students will solve logarithmic expressions.
  (Application) (Mathematics, Algebra II, 6.b)
• The students will solve logarithmic functions.
  (Application) (Mathematics, Algebra II, 6.c)

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John Napier

• Born in Edinburgh, Scotland, in 1550
• Began education at St. Andrews University at the
  age of 13
• Likely acquired mathematical knowledge at the
  University of Paris
• Died April 4, 1617 in Edinburgh, Scotland

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Logarithms

• Definition If b and y are positive where b?1,
  then the logarithm of y with base b (logby) is
  defined as
• logby x
• if and only if bx y.

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Special Logarithms

• logb1 0
• Why? b0 1
• logbb 1
• Why? b1 b
• The logarithm with base 10 is called the common
  logarithm. (log10 or log)
• The logarithm with base e is called the natrual
  logarithm. (loge or ln)

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Examples

• Evaluate the expression log381

• 3x 81

• 3x 34

• x 4

• Evaluate the expression log1/28

• (1/2)x 8

• (1/2)x 23

• x -3

• (1/2)x (1/2)-3

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Logarithmic Functions

• Exponential functions and logarithmic functions
  are inverses
• undo each other
• If g(x) logbx and f(x) bx, then g(f(x))
  logbbx x and
• f(g(x)) blogbx x.

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Examples

• Simplify the expression 10log2

• 10log2 2

• Simplify the expression log39x

• log39x log3(32)x

• log332x

• 2x

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More Examples

• Find the inverse of y log3x

• Use the definition of a logarithm

• y 3x

• Find the inverse of y ln(x 1)

• y ln(x 1)

• x ln(y 1) (switch x and y)

• ex y 1 (write in exponential form)

• ex 1 y (solve for y)

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Assessment

1. Write log7b 13 in exponential form.
2. Write 43 64 in logarithmic form.
3. Solve the equation logx(1/32) -5.
4. Simplify log5252.
5. Evaluate log4256.
6. Find the inverse of y ln(2x 5)

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Closing Questions

• What did we learn about today?
• Can anyone tell me the definition of a logarithm?
• Where might you use logarithms?

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(No Transcript)