3.2 Logarithmic Functions
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Transcript of 3.2 Logarithmic Functions
3.2 Logarithms
• All of the material in this slideshow is important, however the only slides than must be included in your notes are the Recap slides at the end.
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IntroSolving for an
answerSolving for a base Solving for an
exponent
1. 34 = x
81 = x
2. x2 = 49
x = ±7
3. 3x = 729
log3729=3New Calc:
Alpha window
5:logBASE(
Older Calc:
log(729)/log(3)
Answer: 6
b/c 36 = 729
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Now you try…
4. 25 = x 5. x3 = 125 6. 5x = 3125
Plug in calc 32 = x
3 3 3 125
5
x
x
5log 3125
log expbase
x
ans
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Every logarithmic equation has an equivalent exponential form:
y = loga x is equivalent to x = a y
A logarithmic function is the inverse function of an exponential function.
A logarithm is an exponent!
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y = log2( )2
1 = 2 y
2
1
Examples: Write the equivalent exponential equation and solve for y.
1 = 5 yy = log51
16 = 4y y = log416
16 = 2yy = log216
SolutionEquivalent Exponential
Equation
Logarithmic Equation
16 = 24 y = 4
2
1= 2-1 y = –1
16 = 42 y = 2
1 = 50 y = 0
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3.2 Material • Logarithms are used when solving for an exponent.
baseexp =ans logbaseans=exp
• In your calculator: log is base 10 ln is base e
Properties of Logarithms and Ln
1. loga 1 = 0
2. logaa = 1
3. logaax = x
• The graph is the inverse of y=ax (reflected over y =x )
VA: x = 0 , x-intercept (1,0), increasing
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The graphs of logarithmic functions are similar for different values of a. f(x) = loga x
x-intercept (1, 0)
increasing
reflection of y = a x in y = x
VA: x = 0
Graph of f (x) = loga x
x
y y = x
y = log2 x
y = a xy-axis
verticalasymptote
x-intercept(1, 0)
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Example: Graph the common logarithm function f(x) = log10 x.
y
x5
–5
f(x) = log10 x
x = 0 vertical
asymptote
(1, 0) x-intercept
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The function defined by f(x) = loge x = ln x
is called the natural logarithm function.
Use a calculator to evaluate: ln 3, ln –2, ln 100
ln 3ln –2ln 100
Function Value Keystrokes Display
LN 3 ENTER 1.0986122ERRORLN –2 ENTER
LN 100 ENTER 4.6051701
y = ln xy
x5
–5
y = ln x is equivalent to e y = x
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Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
2
1ln
e 2ln 2 e inverse property
eln3 3)1(3 property 2
00 1ln property 1
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Example: The formula (t in years) is used to estimate
the age of organic material. The ratio of carbon 14 to carbon 12
in a piece of charcoal found at an archaeological dig is .
How old is it?
82231210
1 t
eR
To the nearest thousand years the charcoal is 57,000 years old.
original equation158223
12 10
1
10
1
t
e
multiply both sides by 10123
8223
10
1
t
e
take the ln of both sides1000
1lnln 8223
t
e
inverse property1000
1ln
8223
t
56796907.6 82231000
1ln 8223
t
1510
1R
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3.2 Material Recap • Logarithms are used when solving for an exponent.
logab=c ac =b
• In your calculator: log is base 10 ln is base e
Properties of Logarithms and Ln
1. loga 1 = 0 because when written in exponential form a0 = 1
2. logaa = 1 because when written in exponential form a1 = a
3. logaax = x because when written in exponential form ax = ax
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Graph the function f(x) = log x.
y
x5
–5
f(x) = log10 x
x = 0 vertical
asymptote
(1, 0) x-intercept
• The graph is the inverse of y=ax (reflected over y =x )
VA: x = 0 , x-intercept (1,0), increasing
3.2 Material Recap
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Practice ProblemsPage 236 (all work and solutions on blackboard)
•#1–16 pick any 4
•17–22 pick any 3
•27–29 all
•31–37 pick any 4
•45–52 pick any 3
•53–59 pick any 3
•69–71 all
•79–86 pick any 4