8.4 Logarithms

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8.4 Logarithms

• p. 486

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Evaluating Log Expressions

• We know 22 4 and 23 8
• But for what value of y does 2y 6?
• Because 22lt6lt23 you would expect the answer to be
  between 2 3.
• To answer this question exactly, mathematicians
  defined logarithms.

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Definition of Logarithm to base a

• Let a x be positive numbers a ? 1.
• The logarithm of x with base a is denoted by
  logax and is defined
• logax y iff ay x
• This expression is read log base a of x
• The function f(x) logax is the logarithmic
  function with base a.

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• The definition tells you that the equations logax
  y and ay x are equivilant.
• Rewriting forms
• To evaluate log3 9 x ask yourself
• Self 3 to what power is 9?
• 32 9 so log39 2

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Log form Exp. form

• log216 4
• log1010 1
• log31 0
• log10 .1 -1
• log2 6 2.585

• 24 16
• 101 10
• 30 1
• 10-1 .1
• 22.585 6

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Evaluate without a calculator

• 3x 81
• 5x 125
• 4x 256
• 2x (1/32)

• log381
• Log5125
• Log4256
• Log2(1/32)

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-5
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Evaluating logarithms now you try some!
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• Log 4 16
• Log 5 1
• Log 4 2
• Log 3 (-1)
• (Think of the graph of y3x)

0
½ (because 41/2 2)
undefined
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You should learn the following general forms!!!

• Log a 1 0 because a0 1
• Log a a 1 because a1 a
• Log a ax x because ax ax

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Natural logarithms

• log e x ln x
• ln means log base e

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Common logarithms

• log 10 x log x
• Understood base 10 if nothing is there.

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Common logs and natural logs with a calculator
log10 button ln button
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• g(x) log b x is the inverse of
• f(x) bx
• f(g(x)) x and g(f(x)) x
• Exponential and log functions are inverses and
  undo each other

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• So g(f(x)) logbbx x
• f(g(x)) blogbx x
• 10log2
• Log39x
• 10logx
• Log5125x

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Log3(32)x
Log332x
2x
x
3x
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Finding Inverses

• Find the inverse of
• y log3x
• By definition of logarithm, the inverse is y3x
• OR write it in exponential form and switch the x
  y! 3y x 3x y

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Finding Inverses cont.

• Find the inverse of
• Y ln (x 1)
• X ln (y 1) Switch the x y
• ex y 1 Write in exp form
• ex 1 y solve for y

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Assignment
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Graphs of logs

• y logb(x-h)k
• Has vertical asymptote xh
• The domain is xgth, the range is all reals
• If bgt1, the graph moves up to the right
• If 0ltblt1, the graph moves down to the right

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Graph y log1/3x-1

• Plot (1/3,0) (3,-2)
• Vert line x0 is asy.
• Connect the dots

X0
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Graph y log5(x2)

• Plot easy points (-1,0) (3,1)
• Label the asymptote x-2
• Connect the dots using the asymptote.

X-2
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Assignment