M110 CLASS NOTES

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M110 CLASS NOTES

• SECTION 5.2
• LOGARITHMIC FUNCTIONS

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RECALL
EXPONENTIAL GROWTH EXPONENTIAL DECAY
a gt 1 0 lt
a lt 1
1
1
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NOTICE
EXPONENTIAL GROWTH EXPONENTIAL DECAY
a gt 1 0 lt
a lt 1
1
1
Both of these functions are invertible
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The Graphs of f-1(x) for f(x) ax
a gt 1 0 lt
a lt 1
1
1
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Notation for f-1(x) Where f(x) ax
a gt 1 0 lt
a lt 1
1
1
g(x) loga(x)
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PROPERTIES OF THE INVERSE
f-1(f(x)) f-1(ax) x
so f(f-1(x)) x so
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PROPERTIES OF THE INVERSE
y ax
x y
loga(y) x
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EXAMPLES Find the Value

• log39
• log28
• log2(1/2)
• log1/24
• log51

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EXAMPLES Find the Value

• log39 2 (because 32 9)
• log28
• log2(1/2)
• log1/24
• log51

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EXAMPLES Find the Value

• log39 2 (because 32 9)
• log28 3 (because 23 8)
• log2(1/2)
• log1/24
• log51

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EXAMPLES Find the Value

• log39 2 (because 32 9)
• log28 3 (because 23 8)
• log2(1/2) -1 (because 2-1 1/2)
• log1/24
• log51

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EXAMPLES Find the Value

• log39 2 (because 32 9)
• log28 3 (because 23 8)
• log2(1/2) -1 (because 2-1 1/2)
• log1/24 -2 (because (1/2)-2 4)
• log51

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EXAMPLES Find the Value

• log39 2 (because 32 9)
• log28 3 (because 23 8)
• log2(1/2) -1 (because 2-1 1/2)
• log1/24 -2 (because (1/2)-2 4)
• log51 0 (because 50 1)

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EXAMPLES Solve

• log3x -2
• logx10 2
• log2(23) x

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EXAMPLES Solve

• log3x -2 3-2 x 1/32 x x 1/9
• logx10 2
• log2(23) x

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EXAMPLES Solve

• log3x -2 3-2 x 1/32 x x 1/9
• logx10 2 x2 10 x
• log2(23) x

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EXAMPLES Solve

• log3x -2 3-2 x 1/32 x x 1/9
• logx10 2 x2 10 x
• log2(23) x 2x 23 x 3

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NOTATION

• loge x ln x
  The Natural Logarithm
• log10 x log x
  The Common Logarithm

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PROPERTIES USING THE NOTATION

• ln ex x
• log 10x x
• eln x x
• 10log x x

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EXAMPLES Find the domain of

• f(x) log (x 3)
• f(x) ln (2 6x)
• f(x) log5 (x2 2x)

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EXAMPLES Find the domain of

• f(x) log (x 3)
• x 3 gt 0 x gt -3 (-3, ?)
• f(x) ln (2 6x)
• f(x) log5 (x2 2x)

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EXAMPLES Find the domain of

• f(x) log (x 3)
• x 3 gt 0 x gt -3 (-3, ?)
• f(x) ln (2 6x)
• 2 6x gt 0 -6x gt -2 x lt 1/3 (-?,
  1/3)
• f(x) log5 (x2 2x)

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EXAMPLES Find the domain of

• f(x) log (x 3)
• x 3 gt 0 x gt -3 (-3, ?)
• f(x) ln (2 6x)
• 2 6x gt 0 -6x gt -2 x lt 1/3 (-?,
  1/3)
• f(x) log5 (x2 2x)
• x2 2x gt 0 (-?, 0) ? (2, ?)

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End of Section 5.2